Abstract:
The present invention discloses a method for solving the Navier-Stokes equation of blood dynamics as a Non-Newtonian fluid in the left ventricle. The method seeks to provide a model of the model of myocardial motion as an elastic membrane. This invention provides a new method to study the blood flow inside a biological membrane, estimated using quadratic equations.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. provisional patent application Ser. No. 61/345,615, filed May 18, 2010; 61/434,970 filed on Jan. 21, 2011; and 61/434,979 filed on Jan. 21, 2011, which are incorporated herein by reference in their entireties. 
    
    
     FIELD OF INVENTION 
     The present invention relates to a method for studying blood flow regionally near echocardiography samples and globally inside the left ventricle and software and system thereof. 
     BACKGROUND OF INVENTION 
     The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton&#39;s second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity), plus a pressure term. 
     The equations are useful because they describe the physics of many things of academic and economic interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier-Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell&#39;s equations they can be used to model and study magneto-hydrodynamics. 
     Together with supplemental equations (for example, conservation of mass) and well formulated boundary conditions, the Navier-Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agree with real world observations. 
     The Navier-Stokes equations assume that the fluid being studied is a continuum not moving at relativistic velocities. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the continuous fluids modelled by the Navier-Stokes equations. Depending on the Knudsen number of the problem, statistical mechanics or possibly even molecular dynamics may be a more appropriate approach. 
     Time tested formulations exist for common fluid families, but the application of the Navier-Stokes equations to less common families tends to result in very complicated formulations which are an area of current research. For this reason, these equations are usually written for Newtonian fluids. Studying such fluids is “simple” because the viscosity model ends up being linear; truly general models for the flow of other kinds of fluids, such as blood as of 2011, do not exist. 
     Solving the Navier-Stocks equations for an arbitrary fluid is an open problem in mathematics and of course, a very good modelling of such this fluid is strongly related to the membrane where the fluid flows on it. The blood as a complicated and Non-Newtonian fluid through the heart&#39;s chambers and heart&#39;s valves is one of the big challenges among mathematical-, medical-, physical- and computer-sciences. So far a lot of studies of the blood flowing through the heart have been attempted by various simple assumptions. 
     For instance, U.S. Pat. No. 5,537,641, assigned to University of Central Florida Research Foundation, Inc. discloses a method for generating a three-dimensional animation model that stimulates a fluid flow on a three-dimensional graphics display. The said patent does not extend the solution of Navier-Stokes equation to non-Newtonian fluids like blood explicitly. 
     U.S. Pat. No. 6,135,957 assigned to U.S. Philips Corporation describes a method of determining the viscosity and the pressure gradient in a blood vessel, including the acquisition of n≧2 blood speed values, corresponding to the same number of n radii of the blood vessel, determined along a diameter situated in a given axial position, formation of a blood speed vector by means of said n blood speed values, and evaluation of said viscosity and pressure gradient on the basis of a transformation of said blood speed value, including formation of a linear relation which directly links a flow rate vector (y) to the speed derivative vector (h), factorized by the viscosity (μ), and to the pressure gradient vector (σ), and simultaneous evaluation of the two values to be determined for the viscosity (μ) and the pressure gradient (σ) on the basis of said direct equation. The said method, as disclosed in U.S. &#39;957, specifically used to determine blood speed, but seemingly does not disclose a method or system for modelling cardiac condition, specifically left ventricle having a main role in cardiac function based on flow. 
     Hence, the present inventors propose a novel system for solution of Navier-Stokes to model not only the normal blood flow inside the left ventricle but also for the other cavities and valves and model heart diseases. 
     SUMMARY OF INVENTION 
     The invention provides method for studying the blood flows regionally near echocardiography samples and globally inside the left ventricle. 
     In an aspect, according to current invention, the blood flow curves are regionally investigated near the neighbourhoods of echocardiography samples that is, the basal, mid and apical anterior, the basal, mid and apical inferior, and the basal, mid and apical lateral, the basal, mid and apical septum. 
     In another aspect, the flow curves investigated hereinabove are used to model heart diseases using echocardiography. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  shows a general flowchart of this invention. 
         FIG. 2  shows a flowchart where states the basal, mid and apical Anterior as three echocardiography samples in the left ventricle, in their corresponded regions. 
         FIG. 3  shows three rendering of the basal, mid and apical anterior in their corresponded regions. 
         FIG. 4  shows a flowchart that models the blood flow curve near a neighbourhood of the basal Anterior in the myocardium of the left ventricle. 
         FIG. 5  shows the blood flow curve near the corresponded neighbourhood of the basal anterior in the myocardium of the left ventricle where has been rendered at Mathlab software. 
         FIG. 6  shows a flowchart that models the blood flow curve near a neighbourhood of the mid Anterior in the myocardium of the left ventricle. 
         FIG. 7  shows the blood flow curve near the corresponded neighborhood of the mid anterior in the myocardium of the left ventricle where has been rendered at Mathlab software. 
         FIG. 8  shows a flowchart that models the blood flow curve near a neighborhood of the apical Anterior in the myocardium of the left ventricle. 
         FIG. 9  shows the blood flow curve near the corresponded neighborhood of the mid anterior in the myocardium of the left ventricle where has been rendered at Mathlab software. 
         FIG. 10  shows a flowchart where states the basal, mid and apical Inferior as three echocardiography samples in the left ventricle, in their corresponded regions. 
         FIG. 11  shows three rendering of the basal, mid and apical inferior in their corresponded regions at Mathlab software. 
         FIG. 12  shows a flowchart that models the blood flow curve near a neighbourhood of the apical Inferior in the myocardium of the left ventricle. 
         FIG. 13  shows the blood flow curve near the corresponded neighbourhood of the apical Inferior in the myocardium of the left ventricle where has been rendered at Mathlab software. 
         FIG. 14  shows a flowchart that models the blood flow curve near a neighborhood of the mid Inferior in the myocardium of the left ventricle. 
         FIG. 15  shows the blood flow curve near the corresponded neighborhood of the mid Inferior in the myocardium of the left ventricle where has been rendered at Mathlab software. 
         FIG. 16  shows a flowchart that models the blood flow curve near a neighborhood of the basal Inferior in the myocardium of the left ventricle. 
         FIG. 17  shows the blood flow curve near the corresponded neighborhood of the basal Inferior in the myocardium of the left ventricle where has been rendered at Mathlab software. 
         FIG. 18  shows a flowchart where states the basal, mid and apical Lateral as three echocardiography samples in the left ventricle, in their corresponded regions. 
         FIG. 19  shows three rendering of the basal, mid and apical lateral in their corresponded regions at Mathlab software. 
         FIG. 20  shows a flowchart that models the blood flow curve near a neighborhood of the basal Lateral in the myocardium of the left ventricle. 
         FIG. 21  shows the blood flow curve near the corresponded neighborhood of the basal Lateral in the myocardium of the left ventricle where has been rendered at Mathlab software. 
         FIG. 22  shows a flowchart that models the blood flow curve near a neighborhood of the mid Lateral in the myocardium of the left ventricle. 
         FIG. 23  shows the blood flow curve near the corresponded neighborhood of the mid Lateral in the myocardium of the left ventricle where has been rendered at Mathlab software. 
         FIG. 24  shows a flowchart that models the blood flow curve near a neighborhood of the apical Lateral in the myocardium of the left ventricle. 
         FIG. 25  shows the blood flow curve near the corresponded neighborhood of the apical Lateral in the myocardium of the left ventricle where has been rendered at Mathlab software. 
         FIG. 26  shows a flowchart where states the basal, mid and apical Septum as three echocardiography samples in the left ventricle, in their corresponded regions. 
         FIG. 27  shows three rendering of the basal, mid and apical septum in their corresponded regions at Mathlab software. 
         FIG. 28  shows a flowchart that models the blood flow curve near a neighborhood of the apical Septum in the myocardium of the left ventricle. 
         FIG. 29  shows the blood flow curve near the corresponded neighborhood of the apical Septum in the myocardium of the left ventricle where has been rendered at Mathlab software. 
         FIG. 30  shows a flowchart that models the blood flow curve near a neighborhood of the mid Septum in the myocardium of the left ventricle. 
         FIG. 31  shows the blood flow curve near the corresponded neighborhood of the mid Septum in the myocardium of the left ventricle where has been rendered at Mathlab software. 
         FIG. 32  shows a flowchart that models the blood flow curve near a neighborhood of the basal Septum in the myocardium of the left ventricle. 
         FIG. 33  shows the blood flow curve near the corresponded neighborhood of the basal Septum in the myocardium of the left ventricle where has been rendered at Mathlab software. 
         FIG. 34  shows a flowchart where states the blood flow curve inside the left ventricle is regionally made by gluing together blood flow curves that have been modeled near echocardiography samples. 
         FIG. 35  shows the 2-D blood flow curve view in the Mathlab software. 
         FIG. 36  shows the flow of the blood inside the left ventricle related to the other works. 
         FIG. 37  shows two blood flow curves from the left to the right one near a small shant in the Mathlab software. 
         FIG. 38  shows blood flow curves from the left to the right near two shants at VSD in Mathlab software. 
         FIG. 39  the first shows the blood flow curve after an anti-anatomic prosthetic Mitral valve replacement and the second shows the blood flow curves for the natural Mitral valve at the left and after an anatomic prosthetic Mitral valve replacement at the right respectively. 
         FIG. 40  shows the blood flow curve after the best prosthetic Mitral valve replacement. 
     
    
    
     DETAILED DESCRIPTION OF INVENTION 
     The invention will now be described in details with reference to various preferred and optional embodiments to make the invention clear. 
     The present invention describes a method for solving the Navier-Stocks equations of the blood dynamic as a Non-Newtonian fluid in the left ventricle for modeling of the myocardial motion in an elastic membrane. 
     In an embodiment the invention provides modelling of the blood flow curves inside the left ventricle by studying the flow of the blood curves near echocardiography samples i.e. the basal, mid and apical Anterior and the basal, mid and apical Inferior and the basal, mid and apical Lateral and the basal, mid and apical Septum. These samples as the material elastic points in the myocardium of the left ventricle induce mechanical parameters to the viscosity of blood. 
     Invention describes method of formulating and calculating the mechanical parameters of blood, numerically, and then applying Navier-Stocks equations to model the blood flow curve regionally and globally inside the left ventricle. The method is summarized as below 
     a. Calculating mechanical parameters of blood near echocardiography samples; 
     b. calculating the myofiber curve for echocardiography samples of step (a); 
     c. studying “quadratic form” for the curve of step (b) for each echocardiography samples; 
     d. determining the blood flow curve for step (c) for each echocardiography samples and 
     e. integrating the blood flow curves of step (d) for determining blood flow curve for left ventricle globally. 
     In an embodiment of invention,  FIG. 2  illustrates a flowchart giving mathematical signs of the basal Anterior and the mid of Anterior and the apical Anterior in their corresponded regions to obtain good formulizations of the induced mechanical parameters of the blood. 
     Referring to  FIG. 3 , invention further describes geometrical modeling of the basal, mid and apical Anterior using Mathlab software as described below. 
     let ε rr,P     bA   , ε ll,P     bA    and ε cc,P     bA    be strain components of the basal Anterior, P bA  we set
 
γ P     bA   ={each mayocardial sample  X  that ε rr,X ×ε ll,X =ε rr,P     bA   ×ε ll,P     bA    and ε rr,X ×ε ll,X ×ε cc,X =ε rr,P     bA   ×ε ll,P     bA   ×ε cc,P     bA   }
 
and similarly for the mid of Anterior and the apical Anterior would have the following sets;
 
γ P     mA   ={each mayocardial sample  X  that ε rr,X ×ε ll,X =ε rr,P     mA   ×ε ll,P     mA    and ε rr,X ×ε ll,X ×ε cc,X =ε rr,P     mA   ×ε ll,P     mA   ×ε cc,P     mA   }
 
γ P     aA   ={each mayocardial sample  X  that ε rr,X ×ε ll,X =ε rr,P     aA   ×ε ll,P     aA    and ε rr,X ×ε ll,X ×ε cc,X =ε rr,P     aA   ×ε ll,P     aA   ×ε cc,P     aA   }
 
     In fact, γ P     bA′    γ P     mA    and γ P     aA    are those myofiber bands in the myocardium where have been called at  FIG. 2 . 
     In a preferred embodiment Q&#39;s at  FIG. 3  have the following algebraic equations: 
     
       
         
           
             
               
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     By a similar argument we have the algebraic equations in Cartesian coordinate of Q&#39;s for the mid of Anterior and the apical Anterior: 
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     In a preferred embodiment, the invention provides an analytical solution of the Navier-Stocks equations in the region O P     mA    of the apical Inferior.  FIG. 5  shows a rendering of these solutions in the mathlab software. 
       FIG. 4  shows the mechanical parameters of blood which were induced by Q P     mA    in region O P     mA    related to apical inferior. The surface is; 
     
       
         
           
             
               
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     In the region O P     bA   , let φ 1,P     bA   (t), φ 2,P     bA   (t) and φ 3,P     bA   (t) are parameterized forms of the projections of the surface F P     bA    on xy-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       bA 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 bA 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       bA 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 bA 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     bA 
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               bA 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       ll 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     bA   ( t )=φ 1,P     bA   ( t )′/|φ 1,P     bA   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     bA 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             bA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                           ⁢             κ     1   ,     P   bA         ⁡     (   t   )       ·       N     1   ,     P   bA         ⁡     (   t   )         =       ⅆ     T     1   ,     P   bA             ⅆ   s         ;                       κ     1   ,     P   bA         ⁡     (   t   )       =       (       (       (       D     P   bA       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   bA   ′3               ;                 a   1,P     bA   ( t )= S   1,P     bA     ″·T   1,P     bA   ( t )+κ 1,P     bA   ( t )· N   1,P     bA   ( t )
 
 T   2,P     bA   ( t )=φ 2,P     bA   ( t )′/|φ 2,P     bA   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     bA 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             bA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                           ⁢             κ     2   ,     P   bA         ⁡     (   t   )       ·       N     2   ,     P   bA         ⁡     (   t   )         =       ⅆ     T     2   ,     P   bA             ⅆ   s         ;                       κ     2   ,     P   bA         ⁡     (   t   )       =       (       (       (       D     P   bA       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   bA   ′3               ;                 a   2,P     bA   ( t )= S   2,P     bA     ″·T   2,P     bA   ( t )+κ 2,P     bA   ( t )· N   2,P     bA   ( t )
 
 T   3,P     bA   ( t )=φ 3,P     bA   ( t )′/|φ 3,P     bA   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     bA 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             bA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                           ⁢             κ     3   ,     P   bA         ⁡     (   t   )       ·       N     3   ,     P   bA         ⁡     (   t   )         =       ⅆ     T     2   ,     P   bA             ⅆ   s         ;                       κ     3   ,     P   bA         ⁡     (   t   )       =       (       (       (       D     P   bA       -       (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   bA   ′3               ;                 a   3,P     bA   ( t )= S   3,P     bA     ·T   3,P     bA   ( t )+κ 3,P     bA   ( t )· N   3,P     bA   ( t )
 
     If (x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     bA    of the basal anterior and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     bA   , C 2,P     bA    and C 3,P     bA    are the graphs of φ 1,P     bA   (t), φ 2,P     bA   (t) and φ 3,P     bA   (t) respectively then the mechanical parameters of the red blood cells in the region O P     bA    are realized by the following formulas: 
                   v     1   ,     P   bA         ⁡     (   t   )       =       ∫       C   1     ,     P   bA         ⁢           T     1   ,     P   bA         ⁡     (   t   )       ⊗     δ   ⁡     (       x   1     ,     x   2     ,     x   3     ,   t     )         ⁢     ⅆ   t           ;                     n     1   ,     P   bA         ⁡     (   t   )       =       ∫       C   1     ,     P   bA         ⁢           N     1   ,     P   bA         ⁡     (   t   )       ⊗     δ   ⁡     (       x   1     ,     x   2     ,     x   3     ,   t     )         ⁢     ⅆ   t           ;                   a     1   ,     P   bA       RBC     ⁡     (   t   )       =       ∫       C   1     ,     P   bA         ⁢           a     1   ,     P   bA         ⁡     (   t   )       ⊗     δ   ⁡     (       x   1     ,     x   2     ,     x   3     ,   t     )         ⁢     ⅆ   t                           v     2   ,     P   bA         ⁡     (   t   )       =       ∫       C   2     ,     P   bA         ⁢           T     2   ,     P   bA         ⁡     (   t   )       ⊗     δ   ⁡     (       x   1     ,     x   2     ,     x   3     ,   t     )         ⁢     ⅆ   t           ;                     n     2   ,     P   bA         ⁡     (   t   )       =       ∫       C   2     ,     P   bA         ⁢           N     2   ,     P   bA         ⁡     (   t   )       ⊗     δ   ⁡     (       x   1     ,     x   2     ,     x   3     ,   t     )         ⁢     ⅆ   t           ;                   a     2   ,     P   bA       RBC     ⁡     (   t   )       =       ∫       C   2     ,     P   bA         ⁢           a     2   ,     P   bA         ⁡     (   t   )       ⊗     δ   ⁡     (       x   1     ,     x   2     ,     x   3     ,   t     )         ⁢     ⅆ   t                           v     3   ,     P   bA         ⁡     (   t   )       =       ∫       C   3     ,     P   bA         ⁢           T     3   ,     P   bA         ⁡     (   t   )       ⊗     δ   ⁡     (       x   1     ,     x   2     ,     x   3     ,   t     )         ⁢     ⅆ   t           ;                     n     3   ,     P   bA         ⁡     (   t   )       =       ∫       C   3     ,     P   bA         ⁢           N     3   ,     P   bA         ⁡     (   t   )       ⊗     δ   ⁡     (       x   1     ,     x   2     ,     x   3     ,   t     )         ⁢     ⅆ   t           ;                   a     3   ,     P   bA       RBC     ⁡     (   t   )       =       ∫       C   3     ,     P   bA         ⁢           a     3   ,     P   bA         ⁡     (   t   )       ⊗     δ   ⁡     (       x   1     ,     x   2     ,     x   3     ,   t     )         ⁢     ⅆ   t               
the formulas as mentioned hereinabove give analytical solution of the Navier-Stocks equations in the region O P     bA    of the basal Anterior. The invention provides, with reference to  FIG. 5 , provides a rendering of this solution in the mathlab software.
 
     In a preferred embodiment, the invention provides an analytical solution of the Navier-Stocks equations in the region O P     mA    of the apical Inferior.  FIG. 7  shows a rendering of these solutions in the mathlab software. 
       FIG. 6  shows the mechanical parameters of blood which were induced by Q P     mA    in region O P     mA    related to apical inferior. The surface is; 
     
       
         
           
             
               
                 F 
                 
                   P 
                   
                     m 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                 
               
               ⁡ 
               
                 ( 
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     , 
                     
                       y 
                       2 
                     
                     , 
                     
                       y 
                       3 
                     
                   
                   ) 
                 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
               - 
               
                 
                   D 
                   
                     P 
                     
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                   
                 
                 . 
               
             
           
         
       
     
     In the region O P     mA   , let φ 1,P     mA   (t), φ 2,P     mA   (t) and φ 3,P     mA   (t) are parameterized forms of the projections of the surface F P     mA    on xy-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       mA 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 mA 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       mA 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 mA 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     mA 
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               mA 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       ll 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     mA   ( t )=φ 1,P     mA   ( t )′/|φ 1,P     mA   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               A 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                           ⁢             κ     1   ,     P   mA         ⁡     (   t   )       ·       N     1   ,     P   mA         ⁡     (   t   )         =       ⅆ     T     1   ,     P   mA             ⅆ   s         ;                       κ     1   ,     P   mA         ⁡     (   t   )       =       (       (       (       D     P   mA       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   mA   ′3               ;                 a   1,P     mA   ( t )= S   1,P     mA     ″·T   1,P     mA   ( t )+κ 1,P     mA   ( t )· N   1,P     mA   ( t )
 
 T   2,P     mA   ( t )=φ 2,P     mA   ( t )′/|φ 2,P     mA   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               A 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                           ⁢             κ     2   ,     P   mA         ⁡     (   t   )       ·       N     2   ,     P   mA         ⁡     (   t   )         =       ⅆ     T     2   ,     P   mA             ⅆ   s         ;                       κ     2   ,     P   mA         ⁡     (   t   )       =       (       (       (       D     P   mA       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   mA   ′3               ;                 a   2,P     mA   ( t )= S   2,P     mA     ″·T   2,P     mA   ( t )+κ 2,P     mA   ( t )· N   2,P     mA   ( t )
 
 T   3,P     mA   ( t )=φ 3,P     mA   ( t )′/|φ 3,P     mA   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               A 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                           ⁢             κ     3   ,     P   mA         ⁡     (   t   )       ·       N     3   ,     P   mA         ⁡     (   t   )         =       ⅆ     T     2   ,     P   mA             ⅆ   s         ;                       κ     3   ,     P   mA         ⁡     (   t   )       =       (       (       (       D     P   mA       -       (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   mA   ′3               ;                 a   3,P     mA   ( t )= S   3,P     mA     ·T   3,P     mA   ( t )+κ 3,P     mA   ( t )· N   3,P     mA   ( t )
 
     If (x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     mA    of the basal anterior and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     mA   , C 2,P     mA    and C 3,P     mA    are the graphs of φ 1,P     mA   (t), φ 2,P     mA   (t) and φ 3,P     mA   (t) respectively then the mechanical parameters of the red blood cells in the region O P     mA    are realized by the following formulas: 
     
       
         
           
             
               
                 
                   v 
                   
                     1 
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           1 
                           , 
                           
                             P 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               A 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     1 
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           1 
                           , 
                           
                             P 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               A 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   1 
                   , 
                   
                     P 
                     
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     1 
                   
                   , 
                   
                     P 
                     
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         1 
                         , 
                         
                           P 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                           
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     2 
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           2 
                           , 
                           
                             P 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               A 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     2 
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           2 
                           , 
                           
                             P 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               A 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   2 
                   , 
                   
                     P 
                     
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     2 
                   
                   , 
                   
                     P 
                     
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         2 
                         , 
                         
                           P 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                           
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     3 
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           3 
                           , 
                           
                             P 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               A 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     3 
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           3 
                           , 
                           
                             P 
                             
                               m 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               A 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   3 
                   , 
                   
                     P 
                     
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     3 
                   
                   , 
                   
                     P 
                     
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         3 
                         , 
                         
                           P 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                           
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
     
     In a preferred embodiment, the invention provides an analytical solution of the Navier-Stocks equations in the region O P     aA    of the apical Inferior.  FIG. 9  shows a rendering of these solutions in the mathlab software. 
       FIG. 8  shows the mechanical parameters of blood which were induced by Q P     aA    in region O P     aA    related to apical inferior. The surface is; 
     
       
         
           
             
               
                 F 
                 
                   P 
                   aA 
                 
               
               ⁡ 
               
                 ( 
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     , 
                     
                       y 
                       2 
                     
                     , 
                     
                       y 
                       3 
                     
                   
                   ) 
                 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
               - 
               
                 D 
                 
                   P 
                   aA 
                 
               
             
           
         
       
     
     In the region O P     aA   , let φ 1,P     aA   (t), φ 2,P     aA   (t) and φ 3,P     aA   (t) are parameterized forms of the projections of the surface F P     aA    on xy-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 aA 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 aA 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     aA 
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               aA 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       ll 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     aA   ( t )=φ 1,P     aA   ( t )′/|φ 1,P     aA   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     aA 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             aA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                           ⁢             κ     1   ,     P   aA         ⁡     (   t   )       ·       N     1   ,     P   aA         ⁡     (   t   )         =       ⅆ     T     1   ,     P   aA             ⅆ   s         ;                       κ     1   ,     P   aA         ⁡     (   t   )       =       (       (       (       D     P   aA       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   aA   ′3               ;                 a   1,P     aA   ( t )= S   1,P     aA     ″·T   1,P     aA   ( t )+κ 1,P     aA   ( t )· N   1,P     aA   ( t )
 
 T   2,P     aA   ( t )=φ 2,P     aA   ( t )′/|φ 2,P     aA   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     aA 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             aA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     2   ,     P   aA         ⁡     (   t   )       ·       N     2   ,     P   aA         ⁡     (   t   )         =       ⅆ     T     2   ,     P   aA             ⅆ   s         ;                       κ     2   ,     P   aA         ⁡     (   t   )       =       (       (       (       D     P   aA       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   aA               ′     ⁢   3             ;           a   2,P     aA   ( t )= S   2,P     aA     ″·T   2,P     aA   ( t )+κ 2,P     aA   ( t )· N   2,P     aA   ( t )
 
 T   3,P     aA   ( t )=φ 3,P     aA   ( t )′/|φ 3,P     aA   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     aA 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             aA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     3   ,     P   aA         ⁡     (   t   )       ·       N     3   ,     P   aA         ⁡     (   t   )         =       ⅆ     T     2   ,     P   aA             ⅆ   s         ;                       κ     3   ,     P   aA         ⁡     (   t   )       =       (       (       (       D     P   aA       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   aA               ′     ⁢   3             ;           a   3,P     aA   ( t )= S   3,P     aA     ·T   3,P     aA   ( t )+κ 3,P     aA   ( t )· N   3,P     aA   ( t )
 
     If (x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     mA    of the basal anterior and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     aA   , C 2,P     aA    and C 3,P     aA    are the graphs of φ 1,P     aA   (t), φ 2,P     aA   (t) and φ 3,P     aA   (t) respectively then the mechanical parameters of the red blood cells in the region O P     mA    are realized by the following formulas: 
     
       
         
           
             
               
                 
                   v 
                   
                     1 
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           1 
                           , 
                           
                             P 
                             aA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     1 
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           1 
                           , 
                           
                             P 
                             aA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   1 
                   , 
                   
                     P 
                     aA 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     1 
                   
                   , 
                   
                     P 
                     aA 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         1 
                         , 
                         
                           P 
                           aA 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     2 
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           2 
                           , 
                           
                             P 
                             aA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     2 
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           2 
                           , 
                           
                             P 
                             aA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   2 
                   , 
                   
                     P 
                     aA 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     2 
                   
                   , 
                   
                     P 
                     aA 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         2 
                         , 
                         
                           P 
                           aA 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     3 
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           3 
                           , 
                           
                             P 
                             aA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     3 
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       aA 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           3 
                           , 
                           
                             P 
                             aA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   3 
                   , 
                   
                     P 
                     aA 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     3 
                   
                   , 
                   
                     P 
                     aA 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         3 
                         , 
                         
                           P 
                           aA 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
     
     In an embodiment the invention provides mathematical signs of the basal Inferior and the mid of Inferior and the apical Inferior in their corresponded regions to gain good formulizations of the induced mechanical parameters of the blood, as shown in  FIG. 10 . 
     Accordingly, let ε rr,P     bl   , ε ll,P     bl    and ε cc,P     bl    be the strain components of the basal Inferior P bl , then
 
γ P     bl   ={each mayocardial sample  X  that ε rr,X ×ε ll,X =ε rr,P     bl   ×ε ll,P     bl    and ε rr,X ×ε ll,X ×ε cc,X =ε rr,P     bl   ×ε ll,P     bl   ×ε cc,P     bl   }
 
similarly for mid and apical inferior the sets are:
 
γ P     ml   ={each mayocardial sample  X  that ε rr,X ×ε ll,X =ε rr,P     ml   ×ε ll,P     ml    and ε rr,X ×ε ll,X ×ε cc,X =ε rr,P     ml   ×ε ll,P     ml   ×ε cc,P     ml   }
 
γ P     al   ={each mayocardial sample  X  that ε rr,X ×ε ll,X =ε rr,P     al   ×ε ll,P     al    and ε rr,X ×ε ll,X ×ε cc,X =ε rr,P     al   ×ε ll,P     al   ×ε cc,P     al   }
 
γ P     bl   , γ P     ml    and γ P     al    are the myofiber bands illustrated in  FIG. 2 . The Q&#39;s have following values
 
     
       
         
           
             
               
                 Q 
                 
                   P 
                   bl 
                 
               
               : 
               
                 D 
                 
                   P 
                   bl 
                 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         
                           ɛ 
                           rr 
                         
                         
                           
                             P 
                             k 
                           
                           , 
                           
                             P 
                             l 
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           
                             ll 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               k 
                             
                           
                           , 
                           
                             P 
                             l 
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           
                             cc 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               k 
                             
                           
                           , 
                           
                             P 
                             l 
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
             
           
         
       
       
         
           
             
               D 
               
                 P 
                 bl 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         
                           ɛ 
                           rr 
                         
                         
                           
                             P 
                             k 
                           
                           , 
                           
                             P 
                             l 
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   
                     1 
                     , 
                     bl 
                   
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           
                             ll 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               k 
                             
                           
                           , 
                           
                             P 
                             l 
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   
                     2 
                     , 
                     bl 
                   
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           
                             cc 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               P 
                               k 
                             
                           
                           , 
                           
                             P 
                             l 
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   
                     3 
                     , 
                     bl 
                   
                   2 
                 
               
             
           
         
       
     
     Where, P k  and P l  are points belonging to γ P     bl    ∩ O P     bl    and if P bl =(y 1, bl , y 2, bl , y 3, bl ) as Cartesian coordinate 
     Similarly, the Cartesian coordinates for Q&#39;s for mid and apical Inferiors are as follows; 
     For the mid of Anterior: 
                 Q     P     m   ⁢           ⁢   l         :     D     P     m   ⁢           ⁢   l           =         (       ∑     k   ,   l       ⁢         ɛ   rr         P   k     ,     P   l         ⁢     ⅆ   t         )     ·     y   1   2       +       (       ∑     k   ,   l       ⁢       ɛ       ll   ⁢           ⁢     P   k       ,     P   l         ⁢     ⅆ   t         )     ·     y   2   2       +       (       ∑     k   ,   l       ⁢       ɛ       cc   ⁢           ⁢     P   k       ,     P   l         ⁢     ⅆ   t         )     ·     y   3   2                       D     P     m   ⁢           ⁢   l         =         (       ∑     k   ,   l       ⁢         ɛ   rr         P   k     ,     P   l         ⁢     ⅆ   t         )     ·     y     1   ,     m   ⁢           ⁢   l       2       +       (       ∑     k   ,   l       ⁢       ɛ       ll   ⁢           ⁢     P   k       ,     P   l         ⁢     ⅆ   t         )     ·     y     2   ,     m   ⁢           ⁢   l       2       +       (       ∑     k   ,   l       ⁢       ɛ       cc   ⁢           ⁢     P   k       ,     P   l         ⁢     ⅆ   t         )     ·     y     3   ,     m   ⁢           ⁢   l       2               
where, P k  and P l  are points belonging to γ P     ml    ∩ O P     ml    and if P ml =(y 1, ml , y 2, ml , y 3, ml ) as Cartesian coordinate.
 
     For apical Anterior: 
                 Q     P   al       :     D     P   al         =         (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   1   2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   2   2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   3   2                       D     P   al       =         (       ∑       k   .     ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     1   ,   al     2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     2   ,   al     2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     3   ,   al     2               
where, P k  and P l  are points belonging to γ P     al    ∩ O P     al    and if P al =(y 1, al , y 2, al , y 3, al ) as Cartesian coordinate.
 
     In a preferred embodiment, the invention provides an analytical solution of the Navier-Stocks equations in the region O P     al    of the apical Inferior.  FIG. 13  shows a rendering of these solutions in the mathlab software. 
       FIG. 12  shows the mechanical parameters of blood which were induced by Q P     al    in region O P     al    related to apical inferior. The surface is; 
     
       
         
           
             
               
                 F 
                 
                   P 
                   al 
                 
               
               ⁡ 
               
                 ( 
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     , 
                     
                       y 
                       2 
                     
                     , 
                     
                       y 
                       3 
                     
                   
                   ) 
                 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
               - 
               
                 D 
                 
                   P 
                   al 
                 
               
             
           
         
       
     
     In the region O P     al   , let φ 1,P     al   (t), φ 2,P     al   (t) and φ 3,P     al   (t) are parameterized forms of the projections of the surface F P     al    on xy-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 al 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 al 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     al 
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               al 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       ll 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     al   ( t )=φ 1,P     al   ( t )′/|φ 1,P     al   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     al 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     1   ,     P   al         ⁡     (   t   )       ·       N     1   ,     P   al         ⁡     (   t   )         =       ⅆ     T     1   ,     P   al             ⅆ   s         ;                       κ     1   ,     P   al         ⁡     (   t   )       =       (       (       (       D     P   al       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   al               ′     ⁢   3             ;           a   1,P     al   ( t )= S   1,P     al     ″·T   1,P     al   ( t )+κ 1,P     al   ( t )· N   1,P     al   ( t )
 
 T   2,P     al   ( t )=φ 2,P     al   ( t )′/|φ 2,P     al   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     al 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     2   ,     P   al         ⁡     (   t   )       ·       N     2   ,     P   al         ⁡     (   t   )         =       ⅆ     T     2   ,     P   al             ⅆ   s         ;                       κ     2   ,     P   al         ⁡     (   t   )       =       (       (       (       D     P   al       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   al               ′     ⁢   3             ;           a   2,P     al   ( t )= S   2,P     al     ″·T   2,P     al   ( t )+κ 2,P     al   ( t )· N   2,P     al   ( t )
 
 T   3,P     al   ( t )=φ 3,P     al   ( t )′/|φ 3,P     al   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     al 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     3   ,     P   al         ⁡     (   t   )       ·       N     3   ,     P   al         ⁡     (   t   )         =       ⅆ     T     2   ,     P   al             ⅆ   s         ;                       κ     3   ,     P   al         ⁡     (   t   )       =       (       (       (       D     P   al       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   al               ′     ⁢   3             ;           a   3,P     al   ( t )= S   3,P     al     ·T   3,P     al   ( t )+κ 3,P     al   ( t )· N   3,P     al   ( t )
 
(x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     al    of the apical Inferior and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     al   , C 2,P     al    and C 3,P     al    are the graphs of φ 1,P     al   (t), φ 2,P     al   (t) and φ 3,P     al   (t) respectively then the mechanical parameters of the red blood cells in the region O P     al    are calculated by the following formulae:
 
     
       
         
           
             
               
                 
                   v 
                   
                     1 
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           1 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     1 
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           1 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   a 
                   
                     1 
                     , 
                     
                       
                         P 
                         al 
                       
                       ⁢ 
                       l 
                     
                   
                   RBC 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         a 
                         
                           1 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     2 
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           2 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     2 
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           2 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   a 
                   
                     2 
                     , 
                     
                       
                         P 
                         al 
                       
                       ⁢ 
                       l 
                     
                   
                   RBC 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         a 
                         
                           2 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     3 
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           3 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     3 
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       al 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           3 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   3 
                   , 
                   
                     P 
                     al 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     3 
                   
                   , 
                   
                     P 
                     al 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         3 
                         , 
                         
                           P 
                           al 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
     
     In another preferred embodiment the invention provides an analytical solution of the Navier-Stocks equations in the region O P     ml    of the mid Inferior.  FIG. 15  shows a rendering of these solutions in the mathlab software. 
       FIG. 14  shows the mechanical parameters of blood which were induced by Q P     ml    in region O P     ml    related to apical inferior. The surface is; 
     
       
         
           
             
               
                 F 
                 
                   P 
                   ml 
                 
               
               ⁡ 
               
                 ( 
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     , 
                     
                       y 
                       2 
                     
                     , 
                     
                       y 
                       3 
                     
                   
                   ) 
                 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
               - 
               
                 D 
                 
                   P 
                   ml 
                 
               
             
           
         
       
     
     In the region O P     ml   , let φ 1,P     ml   (t), φ 2,P     ml   (t) and φ 3,P     ml   (t) are parameterized forms of the projections of the surface F P     ml    on xy-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 ml 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 ml 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     ml 
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               ml 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       ll 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     ml   ( t )=φ 1,P     ml   ( t )′/|φ 1,P     ml   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     ml 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     1   ,     P   ml         ⁡     (   t   )       ·       N     1   ,     P   ml         ⁡     (   t   )         =       ⅆ     T     1   ,     P   ml             ⅆ   s         ;                       κ     1   ,     P   ml         ⁡     (   t   )       =       (       (       (       D     P   ml       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   ml               ′     ⁢   3             ;           a   1,P     ml   ( t )= S   1,P     ml     ″·T   1,P     ml   ( t )+κ 1,P     ml   ( t )· N   1,P     ml   ( t )
 
 T   2,P     ml   ( t )=φ 2,P     ml   ( t )′/|φ 2,P     ml   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     ml 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     2   ,     P   ml         ⁡     (   t   )       ·       N     2   ,     P   ml         ⁡     (   t   )         =       ⅆ     T     2   ,     P   ml             ⅆ   s         ;                       κ     2   ,     P   ml         ⁡     (   t   )       =       (       (       (       D     P   ml       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   ml               ′     ⁢   3             ;           a   2,P     ml   ( t )= S   2,P     ml     ″·T   2,P     ml   ( t )+κ 2,P     ml   ( t )· N   2,P     ml   ( t )
 
 T   3,P     ml   ( t )=φ 3,P     ml   ( t )′/|φ 3,P     ml   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     ml 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     3   ,     P   ml         ⁡     (   t   )       ·       N     3   ,     P   ml         ⁡     (   t   )         =       ⅆ     T     2   ,     P   ml             ⅆ   s         ;                       κ     3   ,     P   ml         ⁡     (   t   )       =       (       (       (       D     P   ml       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   ml               ′     ⁢   3             ;           a   3,P     ml   ( t )= S   3,P     ml     ·T   3,P     ml   ( t )+κ 3,P     ml   ( t )· N   3,P     ml   ( t )
 
(x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     ml    of the mid Inferior and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     ml   , C 2,P     ml    and C 3,P     ml    are the graphs of φ 1,P     ml   (t), φ 2,P     ml   (t) and φ 3,P     ml   (t) respectively then the mechanical parameters of the red blood cells in the region O P     ml    are calculated by the following formulae:
 
     
       
         
           
             
               
                 
                   v 
                   
                     1 
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           1 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     1 
                     , 
                     
                       · 
                       
                         P 
                         ml 
                       
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           1 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   a 
                   
                     1 
                     , 
                     
                       P 
                       ml 
                     
                   
                   RBC 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         a 
                         
                           1 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     2 
                     , 
                     
                       P 
                       
                         m 
                         ⁢ 
                         l 
                       
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           2 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     2 
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           2 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   a 
                   
                     2 
                     , 
                     
                       P 
                       ml 
                     
                   
                   RBC 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         a 
                         
                           2 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     3 
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           3 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     3 
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       ml 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           3 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   3 
                   , 
                   
                     P 
                     ml 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     3 
                   
                   , 
                   
                     P 
                     ml 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         3 
                         , 
                         
                           P 
                           ml 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
     
     In a preferred embodiment, the invention provides an analytical solution of the Navier-Stocks equations in the region O P     bl    of the basal Inferior.  FIG. 17  shows a rendering of these solutions in the mathlab software. 
       FIG. 16  shows the mechanical parameters of blood which were induced by Q P     bl    in region O P     bl    related to apical inferior. The surface is; 
     
       
         
           
             
               
                 F 
                 
                   P 
                   
                     b 
                     ⁢ 
                     l 
                   
                 
               
               ⁡ 
               
                 ( 
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     , 
                     
                       y 
                       2 
                     
                     , 
                     
                       y 
                       3 
                     
                   
                   ) 
                 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
               - 
               
                 D 
                 
                   P 
                   
                     b 
                     ⁢ 
                     l 
                   
                 
               
             
           
         
       
     
     In the region O P     bl   , let φ 1,P     bl   (t), φ 2,P     bl   (t) and φ 3,P     bl   (t) are parameterized forms of the projections of the surface F P     bl    on xy-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       
                         b 
                         ⁢ 
                         l 
                       
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 
                                   b 
                                   ⁢ 
                                   l 
                                 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       
                         b 
                         ⁢ 
                         l 
                       
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 
                                   b 
                                   ⁢ 
                                   l 
                                 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     
                       b 
                       ⁢ 
                       l 
                     
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               
                                 b 
                                 ⁢ 
                                 l 
                               
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       ll 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     bl   ( t )=φ 1,P     bl   ( t )′/|φ 1,P     bl   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     
                       b 
                       ⁢ 
                       l 
                     
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             
                               b 
                               ⁢ 
                               l 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     1   ,     P     b   ⁢   l           ⁡     (   t   )       ·       N     1   ,     P     b   ⁢   l           ⁡     (   t   )         =       ⅆ     T     1   ,     P     b   ⁢   l               ⅆ   s         ;                       κ     1   ,     P     b   ⁢   l           ⁡     (   t   )       =       (       (       (       D     P     b   ⁢   l         -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P     b   ⁢   l                 ′     ⁢   3             ;           a   1,P     bl   ( t )= S   1,P     bl     ″·T   1,P     bl   ( t )+κ 1,P     bl   ( t )· N   1,P     bl   ( t )
 
 T   2,P     bl   ( t )=φ 2,P     bl   ( t )′/|φ 2,P     bl   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     
                       b 
                       ⁢ 
                       l 
                     
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             
                               b 
                               ⁢ 
                               l 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     2   ,     P     b   ⁢   l           ⁡     (   t   )       ·       N     2   ,     P     b   ⁢   l           ⁡     (   t   )         =       ⅆ     T     2   ,     P     b   ⁢   l               ⅆ   s         ;                       κ     2   ,     P     b   ⁢   l           ⁡     (   t   )       =       (       (       (       D     P     b   ⁢   l         -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P     b   ⁢   l                 ′     ⁢   3             ;           a   2,P     bl   ( t )= S   2,P     bl     ″·T   2,P     bl   ( t )+κ 2,P     bl   ( t )· N   2,P     bl   ( t )
 
 T   3,P     bl   ( t )=φ 3,P     bl   ( t )′/|φ 3,P     bl   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     
                       b 
                       ⁢ 
                       l 
                     
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             
                               b 
                               ⁢ 
                               l 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     3   ,     P     b   ⁢   l           ⁡     (   t   )       ·       N     3   ,     P     b   ⁢   l           ⁡     (   t   )         =       ⅆ     T     2   ,     P     b   ⁢   l               ⅆ   s         ;                       κ     3   ,     P     b   ⁢   l           ⁡     (   t   )       =       (       (       (       D     P     b   ⁢   l         -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P     b   ⁢   l                 ′     ⁢   3             ;           a   3,P     bl   ( t )= S   3,P     bl     ·T   3,P     bl   ( t )+κ 3,P     bl   ( t )· N   3,P     bl   ( t )
 
(x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     bl    of the basal inferior and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     ml   , C 2,P     ml    and C 3,P     ml    are the graphs of φ 1,P     bl   (t), φ 2,P     bl   (t) and φ 3,P     bl   (t) respectively then the mechanical parameters of the red blood cells in the region O P     bl    are calculated by the following formulae:
 
     
       
         
           
             
               
                 
                   v 
                   
                     1 
                     , 
                     
                       P 
                       bl 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       bl 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           1 
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     1 
                     , 
                     
                       P 
                       bl 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       bl 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           1 
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   a 
                   
                     1 
                     , 
                     
                       P 
                       bl 
                     
                   
                   RBC 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       bl 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         a 
                         
                           1 
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         v 
                         
                           2 
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         
                           
                             C 
                             2 
                           
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               T 
                               
                                 2 
                                 , 
                                 
                                   P 
                                   bl 
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           ⊗ 
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                 
                                 , 
                                 
                                   x 
                                   2 
                                 
                                 , 
                                 
                                   x 
                                   3 
                                 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     2 
                     , 
                     
                       P 
                       bl 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       bl 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           2 
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   a 
                   
                     2 
                     , 
                     
                       P 
                       bl 
                     
                   
                   RBC 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       bl 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         a 
                         
                           2 
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         v 
                         
                           3 
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         
                           
                             C 
                             3 
                           
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               T 
                               
                                 3 
                                 , 
                                 
                                   P 
                                   bl 
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           ⊗ 
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                 
                                 , 
                                 
                                   x 
                                   2 
                                 
                                 , 
                                 
                                   x 
                                   3 
                                 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     3 
                     , 
                     
                       P 
                       bl 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       bl 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           3 
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   3 
                   , 
                   
                     P 
                     bl 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     3 
                   
                   , 
                   
                     P 
                     bl 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         3 
                         , 
                         
                           P 
                           bl 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
     
     In an embodiment, as illustrated in  FIG. 18 , the invention provides mathematical signs of basal Lateral, mid Lateral and apical Lateral to obtain good formulizations of the induced mathematical parameters of the blood. 
     The invention further provides geometrical modelling of the basal, mid and apical Lateral as described below; 
     Let ε rr,P     bL   , ε ll,P     bL    and ε cc,P     bL    be the strain components of the basal Inferior P bL , then
 
γ P     bL   ={each mayocardial sample  X  that ε rr,X ×ε ll,X =ε rr,P     bL   ×ε ll,P     bL    and ε rr,X ×ε ll,X ×ε cc,X =ε rr,P     bL   ×ε ll,P     bL   ×ε cc,P     bL   }
 
similarly for mid and apical inferior the sets are:
 
γ P     mL   ={each mayocardial sample  X  that ε rr,X ×ε ll,X =ε rr,P     mL   ×ε ll,P     mL    and ε rr,X ×ε ll,X ×ε cc,X =ε rr,P     mL   ×ε ll,P     mL   ×ε cc,P     mL   }
 
γ P     aL   ={each mayocardial sample  X  that ε rr,X ×ε ll,X =ε rr,P     aL   ×ε ll,P     aL    and ε rr,X ×ε ll,X ×ε cc,X =ε rr,P     aL   ×ε ll,P     aL   ×ε cc,P     aL   }
 
γ P     bL   , γ P     mL    and γ P     aL    are the myofiber bands illustrated in  FIG. 2 . The Q&#39;s have following values
 
     
       
         
           
             
               
                 Q 
                 
                   P 
                   bL 
                 
               
               : 
               
                 D 
                 
                   P 
                   bL 
                 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
             
           
         
       
       
         
           
             
               D 
               
                 P 
                 bL 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   
                     1 
                     , 
                     bL 
                   
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   
                     2 
                     , 
                     bL 
                   
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   
                     3 
                     , 
                     bL 
                   
                   2 
                 
               
             
           
         
       
     
     Where, P k  and P l  are points belonging to γ P     bL    ∩ O P     bL    and if P bL =(y 1, bL , y 2, bL , y 3, bL ) as Cartesian coordinate 
     Similarly, the Cartesian coordinates for Q&#39;s for mid and apical Inferiors are as follows; 
     For the mid of Anterior: 
                 Q     P   mL       :       D   P     bL       =         (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   1   2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   2   2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   3   2                       D     P   mL       =         (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     1   ,   mL     2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     2   ,   mL     2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     3   ,   mL     2               
where, P k  and P l  are points belonging to γ P     mL    ∩ O P     mL    and if P mL =(y 1, mL , y 2, mL , y 3, mL ) as Cartesian coordinate:
 
     For apical Anterior: 
                 Q     P     a   ⁢   L         :       D   P     aL       =         (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   1   2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   2   2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   3   2                       D     P     a   ⁢   L         =         (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     1   ,     a   ⁢   L       2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     2   ,     a   ⁢   L       2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     3   ,     a   ⁢   L       2               
where, P k  and P l  are points belonging to γ P     aL    ∩ O P     aL    and if P aL =(y 1, aL , y 2, aL , y 3, aL ) as Cartesian coordinate.
 
     In a preferred embodiment, the invention provides an analytical solution of the Navier-Stocks equations in the region O P     aL    of the apical Inferior.  FIG. 21  shows a rendering of these solutions in the mathlab software. 
       FIG. 20 , illustrates mechanical parameters of blood induced by Q P     aL    in the region O P     aL    related to apical Lateral. The surface parameters are as follows 
     
       
         
           
             
               
                 F 
                 
                   P 
                   aL 
                 
               
               ⁡ 
               
                 ( 
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     , 
                     
                       y 
                       2 
                     
                     , 
                     
                       y 
                       3 
                     
                   
                   ) 
                 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
               - 
               
                 D 
                 
                   P 
                   aL 
                 
               
             
           
         
       
     
     In the region O P     aL   , let φ 1,P     aL   (t), φ 2,P     aL   (t) and φ 3,P     aL   (t) are parameterized forms of the projections of the surface F P     aL    on xy-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 aL 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 aL 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     aL 
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               aL 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       ll 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     aL   ( t )=φ 1,P     aL   ( t )′/|φ 1,P     aL   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     aL 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     1   ,     P   aL         ⁡     (   t   )       ·       N     1   ,     P   aL         ⁡     (   t   )         =       ⅆ     T     1   ,     P   aL             ⅆ   s         ;                       κ     1   ,     P   aL         ⁡     (   t   )       =       (       (       (       D     P   aL       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   aL       ′3           ;           a   1,P     aL   ( t )= S   1,P     aL     ″·T   1,P     aL   ( t )+κ 1,P     aL   ( t )· N   1,P     aL   ( t )
 
 T   2,P     aL   ( t )=φ 2,P     aL   ( t )′/|φ 2,P     aL   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     aL 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     2   ,     P   aL         ⁡     (   t   )       ·       N     2   ,     P   aL         ⁡     (   t   )         =       ⅆ     T     2   ,     P   aL             ⅆ   s         ;                       κ     2   ,     P   aL         ⁡     (   t   )       =       (       (       (       D     P   aL       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   aL       ′3           ;           a   2,P     aL   ( t )= S   2,P     aL     ″·T   2,P     aL   ( t )+κ 2,P     aL   ( t )· N   2,P     aL   ( t )
 
 T   3,P     aL   ( t )=φ 3,P     aL   ( t )′/|φ 3,P     aL   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     aL 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     3   ,     P   aL         ⁡     (   t   )       ·       N     3   ,     P   aL         ⁡     (   t   )         =       ⅆ     T     2   ,     P   aL             ⅆ   s         ;                       κ     3   ,     P   aL         ⁡     (   t   )       =       (       (       (       D     P   aL       -       (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   aL       ′3           ;           a   3,P     aL   ( t )= S   3,P     aL     ·T   3,P     aL   ( t )+κ 3,P     aL   ( t )· N   3,P     aL   ( t )
 
(x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     aL    of the apical Lateral and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     aL   , C 2,P     aL    and C 3,P     aL    are the graphs of φ 1,P     aL   (t), φ 2,P     aL   (t) and φ 3,P     aL   (t) respectively then the mechanical parameters of the red blood cells in the region O P     aL    are calculated by the following formulae:
 
     
       
         
           
             
               
                 
                   v 
                   
                     1 
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           1 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     1 
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           1 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   a 
                   
                     1 
                     , 
                     
                       P 
                       aL 
                     
                   
                   RBC 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         a 
                         
                           1 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         v 
                         
                           2 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         
                           
                             C 
                             2 
                           
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               T 
                               
                                 2 
                                 , 
                                 
                                   P 
                                   aL 
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           ⊗ 
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                 
                                 , 
                                 
                                   x 
                                   2 
                                 
                                 , 
                                 
                                   x 
                                   3 
                                 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     2 
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           2 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   a 
                   
                     2 
                     , 
                     
                       P 
                       aL 
                     
                   
                   RBC 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         a 
                         
                           2 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         v 
                         
                           3 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         
                           
                             C 
                             3 
                           
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               T 
                               
                                 3 
                                 , 
                                 
                                   P 
                                   aL 
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           ⊗ 
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                 
                                 , 
                                 
                                   x 
                                   2 
                                 
                                 , 
                                 
                                   x 
                                   3 
                                 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     3 
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           3 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   a 
                   
                     3 
                     , 
                     
                       P 
                       aL 
                     
                   
                   RBC 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       aL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         a 
                         
                           3 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
     
     In another preferred embodiment the invention provides an analytical solution of the Navier-Stocks equations in the region O P     mL    of the mid Lateral.  FIG. 23  shows a rendering of these solutions in the mathlab software. 
       FIG. 22  shows the mechanical parameters of blood which were induced by Q P     mL    in region O P     mL    related to apical inferior. The surface is; 
     
       
         
           
             
               
                 F 
                 
                   P 
                   mL 
                 
               
               ⁡ 
               
                 ( 
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     , 
                     
                       y 
                       2 
                     
                     , 
                     
                       y 
                       3 
                     
                   
                   ) 
                 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
               - 
               
                 D 
                 
                   P 
                   mL 
                 
               
             
           
         
       
     
     In the region O P     mL   , let φ 1,P     mL   (t), φ 2,P     mL   (t) and φ 3,P     mL   (t) are parameterized forms of the projections of the surface F P     mL    on xy-axis, xz-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 mL 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 mL 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     mL 
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               mL 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       ll 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     mL   ( t )=φ 1,P     mL   ( t )′/|φ 1,P     mL   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     mL 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     1   ,     P   mL         ⁡     (   t   )       ·       N     1   ,     P   mL         ⁡     (   t   )         =       ⅆ     T     1   ,     P   mL             ⅆ   s         ;                       κ     1   ,     P   mL         ⁡     (   t   )       =       (       (       (       D     P   mL       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   mL       ′3           ;           a   1,P     mL   ( t )= S   1,P     mL     ″·T   1,P     mL   ( t )+κ 1,P     mL   ( t )· N   1,P     mL   ( t )
 
 T   2,P     mL   ( t )=φ 2,P     mL   ( t )′/|φ 2,P     mL   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     mL 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     2   ,     P   mL         ⁡     (   t   )       ·       N     2   ,     P   mL         ⁡     (   t   )         =       ⅆ     T     2   ,     P   mL             ⅆ   s         ;                       κ     2   ,     P   mL         ⁡     (   t   )       =       (       (       (       D     P   mL       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   mL       ′3           ;           a   2,P     mL   ( t )= S   2,P     mL     ″·T   2,P     mL   ( t )+κ 2,P     mL   ( t )· N   2,P     mL   ( t )
 
 T   3,P     mL   ( t )=φ 3,P     mL   ( t )′/|φ 3,P     mL   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     mL 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     3   ,     P   mL         ⁡     (   t   )       ·       N     3   ,     P   mL         ⁡     (   t   )         =       ⅆ     T     2   ,     P   mL             ⅆ   s         ;                       κ     3   ,     P   mL         ⁡     (   t   )       =       (       (       (       D     P   mL       -       (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   mL       ′3           ;           a   3,P     mL   ( t )= S   3,P     mL     ·T   3,P     mL   ( t )+κ 3,P     mL   ( t )· N   3,P     mL   ( t )
 
(x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     mL    of the mid Lateral and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     mL   , C 2,P     mL    and C 3,P     mL    are the graphs of φ 1,P     mL   (t), φ 2,P     mL   (t) and φ 3,P     mL   (t) respectively then the mechanical parameters of the red blood cells in the region O P     mL    are calculated by the following formulae:
 
     
       
         
           
             
               
                 
                   v 
                   
                     1 
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           1 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     1 
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           1 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   a 
                   
                     1 
                     , 
                     
                       P 
                       mL 
                     
                   
                   RBC 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         a 
                         
                           1 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         v 
                         
                           2 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         
                           
                             C 
                             2 
                           
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               T 
                               
                                 2 
                                 , 
                                 
                                   P 
                                   mL 
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           ⊗ 
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                 
                                 , 
                                 
                                   x 
                                   2 
                                 
                                 , 
                                 
                                   x 
                                   3 
                                 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     2 
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           2 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   a 
                   
                     2 
                     , 
                     
                       P 
                       mL 
                     
                   
                   RBC 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         a 
                         
                           2 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         v 
                         
                           3 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         
                           
                             C 
                             3 
                           
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               T 
                               
                                 3 
                                 , 
                                 
                                   P 
                                   mL 
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           ⊗ 
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                 
                                 , 
                                 
                                   x 
                                   2 
                                 
                                 , 
                                 
                                   x 
                                   3 
                                 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     3 
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           3 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   a 
                   
                     3 
                     , 
                     
                       P 
                       mL 
                     
                   
                   RBC 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       mL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         a 
                         
                           3 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
     
     In another preferred embodiment the invention provides an analytical solution of the Navier-Stocks equations in the region O P     bL    of the basal Lateral.  FIG. 25  shows a rendering of these solutions in the mathlab software. 
       FIG. 24  shows the mechanical parameters of blood which were induced by Q P     bL    in region O P     bL    related to apical inferior. The surface is; 
     
       
         
           
             
               
                 F 
                 
                   P 
                   
                     b 
                     ⁢ 
                     L 
                   
                 
               
               ⁡ 
               
                 ( 
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     , 
                     
                       y 
                       2 
                     
                     , 
                     
                       y 
                       3 
                     
                   
                   ) 
                 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
               - 
               
                 D 
                 
                   P 
                   
                     b 
                     ⁢ 
                     L 
                   
                 
               
             
           
         
       
     
     In the region O P     bL   , let φ 1,P     bL   (t), φ 2,P     bL   (t) and φ 3,P     bL   (t) are parameterized forms of the projections of the surface F P     bL    on xy-axis, xz-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 bL 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 bL 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     bL 
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               bL 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       rr 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     bL   ( t )=φ 1,P     bL   ( t )′/|φ 1,P     bL   ( t )′|;
 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         S 
                         
                           1 
                           , 
                           
                             P 
                             bL 
                           
                         
                       
                       = 
                       
                         
                           ∫ 
                           
                             t 
                             o 
                           
                           t 
                         
                         ⁢ 
                         
                           
                             
                               
                                 φ 
                                 
                                   1 
                                   , 
                                   
                                     P 
                                     bL 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 u 
                                 ) 
                               
                             
                             ′ 
                           
                           ⁢ 
                           
                             ⅆ 
                             u 
                           
                         
                       
                     
                     ; 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
                     ⁢             κ     1   ,     P   bL         ⁡     (   t   )       ·       N     1   ,     P   bL         ⁡     (   t   )         =       ⅆ     T     1   ,     P   bL             ⅆ   s         ;                       κ     1   ,     P   bL         ⁡     (   t   )       =       (       (       (       D     P   bL       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   bL       ′3           ;           a   1,P     bL   ( t )= S   1,P     bL     ″·T   1,P     bL   ( t )+κ 1,P     bL   ( t )· N   1,P     bL   ( t )
 
 T   2,P     bL   ( t )=φ 2,P     bL   ( t )′/|φ 2,P     bL   ( t )′|;
 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         S 
                         
                           2 
                           , 
                           
                             P 
                             bL 
                           
                         
                       
                       = 
                       
                         
                           ∫ 
                           
                             t 
                             o 
                           
                           t 
                         
                         ⁢ 
                         
                           
                             
                               
                                 φ 
                                 
                                   2 
                                   , 
                                   
                                     P 
                                     bL 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 u 
                                 ) 
                               
                             
                             ′ 
                           
                           ⁢ 
                           
                             ⅆ 
                             u 
                           
                         
                       
                     
                     ; 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
                     ⁢             κ     2   ,     P   bL         ⁡     (   t   )       ·       N     2   ,     P   bL         ⁡     (   t   )         =       ⅆ     T     2   ,     P   bL             ⅆ   s         ;                       κ     2   ,     P   bL         ⁡     (   t   )       =       (       (       (       D     P   bL       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   bL       ′3           ;           a   2,P     bL   ( t )= S   2,P     bL     ″·T   2,P     bL   ( t )+κ 2,P     bL   ( t )· N   2,P     bL   ( t )
 
 T   3,P     bL   ( t )=φ 3,P     bL   ( t )′/|φ 3,P     bL   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     bL 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             bL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     3   ,     P   bL         ⁡     (   t   )       ·       N     3   ,     P   bL         ⁡     (   t   )         =       ⅆ     T     2   ,     P   bL             ⅆ   s         ;                       κ     3   ,     P   bL         ⁡     (   t   )       =       (       (       (       D     P   bL       -       (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   bL       ′3           ;           a   3,P     bL   ( t )= S   3,P     bL     ·T   3,P     bL   ( t )+κ 3,P     bL   ( t )· N   3,P     bL   ( t )
 
(x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     bL    of the basal Lateral and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     bL   , C 2,P     bL    and C 3,P     bL    are the graphs of φ 1,P     bL   (t), φ 2,P     bL   (t) and φ 3,P     bL   (t) respectively then the mechanical parameters of the red blood cells in the region O P     bL    are calculated by the following formulae:
 
     
       
         
           
             
               
                 
                   v 
                   
                     1 
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           1 
                           , 
                           
                             P 
                             bL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     1 
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           1 
                           , 
                           
                             P 
                             bL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   1 
                   , 
                   
                     P 
                     bL 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     1 
                   
                   , 
                   
                     P 
                     bL 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         1 
                         , 
                         
                           P 
                           bL 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     2 
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           2 
                           , 
                           
                             P 
                             bL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     2 
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           2 
                           , 
                           
                             P 
                             bL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   2 
                   , 
                   
                     P 
                     bL 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     2 
                   
                   , 
                   
                     P 
                     bL 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         2 
                         , 
                         
                           P 
                           bL 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     3 
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           3 
                           , 
                           
                             P 
                             bL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     3 
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       bL 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           3 
                           , 
                           
                             P 
                             bL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   3 
                   , 
                   
                     P 
                     bL 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     3 
                   
                   , 
                   
                     P 
                     bL 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         3 
                         , 
                         
                           P 
                           bL 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
     
     In an embodiment, as illustrated in  FIG. 26 , the invention provides mathematical signs of basal Septum, mid Septum and apical Septum to obtain good formulizations of the induced mathematical parameters of the blood. 
     The invention further provides geometrical modelling of the basal, mid and apical Septum as described below; 
     Let ε rr,P     bS   , ε P     bS    and ε cc,P     bS    be the strain components of the basal Inferior P bS , then
 
γ P     bS   ={each mayocardial sample  X  that ε rr,X ×ε ll,X =ε rr,P     bS   ×ε ll,P     bS    and ε rr,X ×ε ll,X ×ε cc,X =ε rr,P     bS   ×ε ll,P     bS   ×ε cc,P     bS   }
 
similarly for mid and apical inferior the sets are:
 
γ P     mS   ={each mayocardial sample  X  that ε rr,X ×ε ll,X =ε rr,P     mS   ×ε ll,P     mS    and ε rr,X ×ε ll,X ×ε cc,X =ε rr,P     mS   ×ε ll,P     mS   ×ε cc,P     mS   }
 
γ P     aS   ={each mayocardial sample  X  that ε rr,X ×ε ll,X =ε rr,P     aS   ×ε ll,P     aS    and ε rr,X ×ε ll,X ×ε cc,X =ε rr,P     aS   ×ε ll,P     aS   ×ε cc,P     aS   }
 
γ P     bS   , γ P     mS    and γ P     aS    are the myofiber bands illustrated in  FIG. 2 . The Q&#39;s have following values
 
     
       
         
           
             
               
                 Q 
                 
                   P 
                   bS 
                 
               
               ∷ 
               
                 D 
                 
                   P 
                   bS 
                 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
             
           
         
       
       
         
           
             
               D 
               
                 P 
                 bS 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   
                     1 
                     , 
                     bS 
                   
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   
                     2 
                     , 
                     bS 
                   
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   
                     3 
                     , 
                     bS 
                   
                   2 
                 
               
             
           
         
       
     
     Where, P k  and P l  are points belonging to γ P     bS    ∩ O P     bS    and if P bS =(y 1, bS , y 2, bS , y 3, bS ) as Cartesian coordinate 
     Similarly, the Cartesian coordinates for Q&#39;s for mid and apical Inferiors are as follows; 
     For the mid of Septum: 
                 Q     P   mS       ∷     D     P   mS         =         (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   1   2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   2   2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   3   2                         D     P   mS       =         (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     1   ,   mS     2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     2   ,   mS     2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     3   ,   mS     2           ⁢     
     ⁢                 
where, P k  and P l  are points belonging to γ P     mS    ∩ O P     mS    and if P mS =(y 1, mS , y 2, mS , y 3, mS ) as Cartesian coordinate.
 
     For apical Anterior: 
                 Q     P   aS       ∷     D     P   aS         =         (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   1   2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   2   2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y   3   2                       D     P   aS       =         (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     1   ,   aS     2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     2   ,   aS     2       +       (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l           ⁢     ⅆ   t         )     ·     y     3   ,   aS     2               
where, P k  and P l  are points belonging to γ P     aS    ∩ O P     aS    and if P aS =(y 1,aS , y 2,aS , y 3,aS ) as Cartesian coordinate.
 
     In a preferred embodiment, the invention provides an analytical solution of the Navier-Stocks equations in the region O P     aS    of the apical Septum.  FIG. 30  shows a rendering of these solutions in the mathlab software. 
       FIG. 29 , illustrates mechanical parameters of blood induced by Q P     aS    in the region O P     aS    related to apical Lateral. The surface parameters are as follows 
     
       
         
           
             
               
                 F 
                 
                   P 
                   aS 
                 
               
               ⁡ 
               
                 ( 
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     , 
                     
                       y 
                       2 
                     
                     , 
                     
                       y 
                       3 
                     
                   
                   ) 
                 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
               - 
               
                 D 
                 
                   P 
                   aS 
                 
               
             
           
         
       
     
     In the region O P     aS   , let φ 1,P     aS   (t), φ 2,P     aS   (t) and φ 3,P     aS   (t) are parameterized forms of the projections of the surface F P     aS    on xy-axis, xz-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 aS 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 aS 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     aS 
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               aS 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       ll 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     aS   ( t )=φ 1,P     aS   ( t )′/|φ 1,P     aS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     aS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     1   ,     P   aS         ⁡     (   t   )       ·       N     1   ,     P   aS         ⁡     (   t   )         =       ⅆ     T     1   ,     P   aS             ⅆ   s         ;                       κ     1   ,     P   aS         ⁡     (   t   )       =       (       (       (       D     P   aS       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   aS       ′3           ;           a   1,P     aS   ( t )= S   1,P     aS     ″·T   1,P     aS   ( t )+κ 1,P     aS   ( t )· N   1,P     aS   ( t )
 
 T   2,P     aS   ( t )=φ 2,P     aS   ( t )′/|φ 2,P     aS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     aS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     2   ,     P   aS         ⁡     (   t   )       ·       N     2   ,     P   aS         ⁡     (   t   )         =       ⅆ     T     2   ,     P   aS             ⅆ   s         ;                       κ     2   ,     P   aS         ⁡     (   t   )       =       (       (       (       D     P   aS       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   aS       ′3           ;           a   2,P     aS   ( t )= S   2,P     aS     ″·T   2,P     aS   ( t )+κ 2,P     aS   ( t )· N   2,P     aS   ( t )
 
 T   3,P     aS   ( t )=φ 3,P     aS   ( t )′/|φ 3,P     aS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     aS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     3   ,     P   aS         ⁡     (   t   )       ·       N     3   ,     P   aS         ⁡     (   t   )         =       ⅆ     T     2   ,     P   aS             ⅆ   s         ;                       κ     3   ,     P   aS         ⁡     (   t   )       =       (       (       (       D     P   aS       -       (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   aS       ′3           ;           a   3,P     aS   ( t )= S   3,P     aS     ·T   3,P     aS   ( t )+κ 3,P     aS   ( t )· N   3,P     aS   ( t )
 
(x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     aS    of the apical Septum and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     aS   , C 2,P     aS    and C 3,P     aS    are the graphs of φ 1,P     aS   (t), φ 2,P     aS   (t) and φ 3,P     aS   (t) respectively then the mechanical parameters of the red blood cells in the region O P     aS    are calculated by the following formulae:
 
     
       
         
           
             
               
                 
                   v 
                   
                     1 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           1 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     1 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           1 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   1 
                   , 
                   
                     P 
                     aS 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     1 
                   
                   , 
                   
                     P 
                     aS 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         1 
                         , 
                         
                           P 
                           aS 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     2 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           2 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     2 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           2 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   2 
                   , 
                   
                     P 
                     aS 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     2 
                   
                   , 
                   
                     P 
                     aS 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         2 
                         , 
                         
                           P 
                           aS 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     3 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           3 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     3 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           3 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   3 
                   , 
                   
                     P 
                     aS 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     3 
                   
                   , 
                   
                     P 
                     aS 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         3 
                         , 
                         
                           P 
                           aS 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
     
     In a preferred embodiment, the invention provides an analytical solution of the Navier-Stocks equations in the region O P     mS    of the mid Septum.  FIG. 32  shows a rendering of these solutions in the mathlab software. 
       FIG. 31 , illustrates mechanical parameters of blood induced by Q P     mS    in the region O P     mS    related to mid Lateral. The surface parameters are as follows 
     
       
         
           
             
               
                 F 
                 
                   P 
                   mS 
                 
               
               ⁡ 
               
                 ( 
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     , 
                     
                       y 
                       2 
                     
                     , 
                     
                       y 
                       3 
                     
                   
                   ) 
                 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
               - 
               
                 D 
                 
                   P 
                   mS 
                 
               
             
           
         
       
     
     In the region O P     mS   , let φ 1,P     mS   (t), φ 2,P     mS   (t) and φ 3,P     mS   (t) are parameterized forms of the projections of the surface F P     mS    on xy-axis, xz-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 mS 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 mS 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     mS 
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               mS 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       ll 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     mS   ( t )=φ 1,P     mS   ( t )′/|φ 1,P     mS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     mS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     1   ,     P   mS         ⁡     (   t   )       ·       N     1   ,     P   mS         ⁡     (   t   )         =       ⅆ     T     1   ,     P   mS             ⅆ   s         ;                       κ     2   ,     P   mS         ⁡     (   t   )       =       (       (       (       D     P   mS       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   mS         ′   ⁢           ⁢   3             ;           a   1,P     mS   ( t )= S   1,P     mS     ″·T   1,P     mS   ( t )+κ 1,P     mS   ( t )· N   1,P     mS   ( t )
 
 T   2,P     mS   ( t )=φ 2,P     mS   ( t )′/|φ 1,P     mS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     mS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     2   ,     P   mS         ⁡     (   t   )       ·       N     2   ,     P   mS         ⁡     (   t   )         =       ⅆ     T     2   ,     P   mS             ⅆ   s         ;                       κ     2   ,     P   mS         ⁡     (   t   )       =       (       (       (       D     P   mS       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   mS         ′   ⁢           ⁢   3             ;           a   2,P     mS   ( t )= S   2,P     mS     ″·T   2,P     mS   ( t )+κ 2,P     mS   ( t )· N   2,P     mS   ( t )
 
 T   3,P     mS   ( t )=φ 3,P     mS   ( t )′/|φ 3,P     mS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     mS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     3   ,     P   mS         ⁡     (   t   )       ·       N     3   ,     P   mS         ⁡     (   t   )         =       ⅆ     T     2   ,     P   mS             ⅆ   s         ;                       κ     3   ,     P   mS         ⁡     (   t   )       =       (       (       (       D     P   mS       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   mS         ′   ⁢           ⁢   3             ;           a   3,P     mS   ( t )= S   3,P     mS     ·T   3,P     mS   ( t )+κ 3,P     mS   ( t )· N   3,P     mS   ( t )
 
     Following formulae were set;
 
 T   1,P     mS   ( t )=φ 1,P     mS   ( t )′/|φ 3,P     mS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     mS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     1   ,     P   mS         ⁡     (   t   )       ·       N     1   ,     P   mS         ⁡     (   t   )         =       ⅆ     T     1   ,     P   mS             ⅆ   s         ;                       κ     1   ,     P   mS         ⁢     (   t   )       =       (       (       (       D     P   mS       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   mS         ′   ⁢           ⁢   3             ;           a   1,P     mS   ( t )= S   1,P     mS     ″·T   1,P     mS   ( t )+κ 1,P     mS   ( t )· N   1,P     mS   ( t )
 
 T   2,P     mS   ( t )=φ 2,P     mS   ( t )′/|φ 2,P     mS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     mS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     2   ,     P   mS         ⁡     (   t   )       ·       N     2   ,     P   mS         ⁡     (   t   )         =       ⅆ     T     2   ,     P   mS             ⅆ   s         ;                       κ     2   ,     P   mS         ⁡     (   t   )       =       (       (       (       D     P   mS       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   mS         ′   ⁢           ⁢   3             ;           a   2,P     mS   ( t )= S   2,P     mS     ″·T   2,P     mS   ( t )+κ 2,P     mS   ( t )· N   2,P     mS   ( t )
 
 T   3,P     mS   ( t )=φ 3,P     mS   ( t )′/|φ 3,P     mS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     mS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     3   ,     P   mS         ⁡     (   t   )       ·       N     3   ,     P   mS         ⁡     (   t   )         =       ⅆ     T     2   ,     P   mS             ⅆ   s         ;                       κ     3   ,     P   mS         ⁢     (   t   )       =       (       (       (       D     P   mS       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   mS         ′   ⁢           ⁢   3             ;           a   3,P     mS   ( t )= S   3,P     mS     ·T   3,P     mS   ( t )+κ 3,P     mS   ( t )· N   3,P     mS   ( t )
 
(x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     mS    of the apical Septum and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     mS   , C 2,P     mS    and C 3,P     mS    are the graphs of φ 1,P     mS   (t), φ 2,P     mS   (t) and φ 3,P     mS   (t) respectively then the mechanical parameters of the red blood cells in the region O P     mS    are calculated by the following formulae:
 
     
       
         
           
             
               
                 
                   v 
                   
                     1 
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           1 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     1 
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           1 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   1 
                   , 
                   
                     P 
                     mS 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     1 
                   
                   , 
                   
                     P 
                     mS 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         1 
                         , 
                         
                           P 
                           mS 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     2 
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           2 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     2 
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           2 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   2 
                   , 
                   
                     P 
                     mS 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     2 
                   
                   , 
                   
                     P 
                     mS 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         2 
                         , 
                         
                           P 
                           mS 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     3 
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           3 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     3 
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       mS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           3 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   3 
                   , 
                   
                     P 
                     mS 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     3 
                   
                   , 
                   
                     P 
                     mS 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         3 
                         , 
                         
                           P 
                           mS 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
     
     In a preferred embodiment, the invention provides an analytical solution of the Navier-Stocks equations in the region O P     aS    of the apical Septum.  FIG. 30  shows a rendering of these solutions in the mathlab software. 
       FIG. 29 , illustrates mechanical parameters of blood induced by Q P     aS    in the region O P     aS    related to apical Lateral. The surface parameters are as follows 
     
       
         
           
             
               
                 F 
                 
                   P 
                   aS 
                 
               
               ⁡ 
               
                 ( 
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     , 
                     
                       y 
                       2 
                     
                     , 
                     
                       y 
                       3 
                     
                   
                   ) 
                 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
               - 
               
                 D 
                 
                   P 
                   aS 
                 
               
             
           
         
       
     
     In the region O P     aS   , let φ 1,P     aS   (t), φ 2,P     aS   (t) and φ 3,P     aS   (t) are parameterized forms of the projections of the surface F P     aS    on xy-axis, xz-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 aS 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 aS 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     aS 
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               aS 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       ll 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     aS   ( t )=φ 1,P     aS   ( t )′/|φ 1,P     aS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     aS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     1   ,     P   aS         ⁡     (   t   )       ·       N     1   ,     P   aS         ⁡     (   t   )         =       ⅆ     T     1   ,     P   aS             ⅆ   s         ;                       κ     1   ,     P   aS         ⁢     (   t   )       =       (       (       (       D     P   aS       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   aS         ′   ⁢           ⁢   3             ;           a   1,P     aS   ( t )= S   1,P     aS     ·T   1,P     aS   ( t )+κ 1,P     aS   ( t )· N   1,P     aS   ( t )
 
 T   2,P     aS   ( t )=φ 2,P     aS   ( t )′/|φ 2,P     aS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     aS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     2   ,     P   aS         ⁡     (   t   )       ·       N     2   ,     P   aS         ⁡     (   t   )         =       ⅆ     T     2   ,     P   aS             ⅆ   s         ;                       κ     2   ,     P   aS         ⁡     (   t   )       =       (       (       (       D     P   aS       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   aS         ′   ⁢           ⁢   3             ;           a   2,P     aS   ( t )= S   2,P     aS     ″·T   2,P     aS   ( t )+κ 2,P     aS   ( t )· N   2,P     aS   ( t )
 
 T   3,P     aS   ( t )=φ 3,P     aS   ( t )′/|φ 3,P     aS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     aS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     3   ,     P   aS         ⁡     (   t   )       ·       N     3   ,     P   aS         ⁡     (   t   )         =       ⅆ     T     2   ,     P   aS             ⅆ   s         ;                       κ     3   ,     P   aS         ⁢     (   t   )       =       (       (       (       D     P   aS       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   aS         ′   ⁢           ⁢   3             ;           a   3,P     aS   ( t )= S   3,P     aS     ·T   3,P     aS   ( t )+κ 3,P     aS   ( t )· N   3,P     aS   ( t )
 
(x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     aS    of the apical Septum and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     aS   , C 2,P     aS    and C 3,P     aS    are the graphs of φ 1,P     aS   (t), φ 2,P     aS   (t) and φ 3,P     aS   (t) respectively then the mechanical parameters of the red blood cells in the region O P     aS    are calculated by the following formulae:
 
     
       
         
           
             
               
                 
                   v 
                   
                     1 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           1 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     1 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           1 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   1 
                   , 
                   
                     P 
                     aS 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     1 
                   
                   , 
                   
                     P 
                     aS 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         1 
                         , 
                         
                           P 
                           aS 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     2 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           2 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     2 
                     ⁢ 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           2 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   2 
                   , 
                   
                     P 
                     aS 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     2 
                   
                   , 
                   
                     P 
                     aS 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         2 
                         , 
                         
                           P 
                           aS 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     3 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           3 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     3 
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       aS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           3 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   3 
                   , 
                   
                     P 
                     aS 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     3 
                   
                   , 
                   
                     P 
                     aS 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         3 
                         ⁢ 
                         
                           P 
                           aS 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
     
     In a preferred embodiment, the invention provides an analytical solution of the Navier-Stocks equations in the region O P     bS    of the base Septum.  FIG. 34  shows a rendering of these solutions in the mathlab software. 
       FIG. 33 , illustrates mechanical parameters of blood induced by Q P     bS    in the region O P     bS    related to mid Lateral. The surface parameters are as follows 
     
       
         
           
             
               
                 F 
                 
                   P 
                   bS 
                 
               
               ⁡ 
               
                 ( 
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     , 
                     
                       y 
                       2 
                     
                     , 
                     
                       y 
                       3 
                     
                   
                   ) 
                 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           rr 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   1 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           ll 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   2 
                   2 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         
                           cc 
                           
                             
                               P 
                               k 
                             
                             , 
                             
                               P 
                               l 
                             
                           
                         
                         ′ 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   y 
                   3 
                   2 
                 
               
               - 
               
                 D 
                 
                   P 
                   bS 
                 
               
             
           
         
       
     
     In the region O P     bS   , let φ 1,P     bS   (t), φ 2,P     bS   (t) and φ 3,P     bS   (t) are parameterized forms of the projections of the surface F P     bS    on xy-axis, xz-axis and yz-axis: 
     
       
         
           
             
               
                 
                   φ 
                   
                     1 
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 bS 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   ll 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   φ 
                   
                     2 
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 ( 
                 
                   t 
                   , 
                   
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               D 
                               
                                 P 
                                 mS 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       , 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ɛ 
                                       
                                         rr 
                                         
                                           
                                             P 
                                             k 
                                           
                                           , 
                                           
                                             P 
                                             l 
                                           
                                         
                                       
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       ⅆ 
                                       t 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         / 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 k 
                                 , 
                                 l 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   cc 
                                   
                                     
                                       P 
                                       k 
                                     
                                     , 
                                     
                                       P 
                                       l 
                                     
                                   
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       1 
                       2 
                     
                   
                 
                 ) 
               
             
             ; 
           
         
       
       
         
           
             
               
                 φ 
                 
                   3 
                   , 
                   
                     P 
                     bS 
                   
                 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             D 
                             
                               P 
                               bS 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     , 
                                     l 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     ɛ 
                                     
                                       ll 
                                       
                                         
                                           P 
                                           k 
                                         
                                         , 
                                         
                                           P 
                                           l 
                                         
                                       
                                     
                                     ′ 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               t 
                               2 
                             
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               , 
                               l 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ɛ 
                               
                                 cc 
                                 
                                   
                                     P 
                                     k 
                                   
                                   , 
                                   
                                     P 
                                     l 
                                   
                                 
                               
                               ′ 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Following formulae were set;
 
 T   1,P     bS   ( t )=φ 1,P     bS   ( t )′/|φ 1,P     bS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     bS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     1   ,     P   bS         ⁡     (   t   )       ·       N     1   ,     P   bS         ⁡     (   t   )         =       ⅆ     T     1   ,     P   bS             ⅆ   s         ;                       κ     1   ,     P   bS         ⁢     (   t   )       =       (       (       (       D     P   bS       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   bS         ′   ⁢           ⁢   3             ;           a   1,P     bS   ( t )= S   1,P     bS     ″·T   1,P     bS   ( t )+κ 1,P     bS   ( t )· N   1,P     bS   ( t )
 
 T   2,P     bS   ( t )=φ 2,P     bS   ( t )′/|φ 2,P     bS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     bS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     2   ,     P   bS         ⁡     (   t   )       ·       N     2   ,     P   bS         ⁡     (   t   )         =       ⅆ     T     2   ,     P   bS             ⅆ   s         ;                       κ     2   ,     P   bS         ⁡     (   t   )       =       (       (       (       D     P   bS       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   bS         ′   ⁢           ⁢   3             ;           a   2,P     bS   ( t )= S   2,P     bS     ″·T   2,P     bS   ( t )+κ 2,P     bS   ( t )· N   2,P     bS   ( t )
 
 T   3,P     bS   ( t )=φ 3,P     bS   ( t )′/|φ 3,P     bS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     bS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                     ⁢             κ     3   ,     P   bS         ⁡     (   t   )       ·       N     3   ,     P   bS         ⁡     (   t   )         =       ⅆ     T     2   ,     P   bS             ⅆ   s         ;                       κ     3   ,     P   bS         ⁢     (   t   )       =       (       (       (       D     P   bS       -       (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢           ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   bS         ′   ⁢           ⁢   3             ;           a   3,P     bS   ( t )= S   3,P     bS     ·T   3,P     bS   ( t )+κ 3,P     bS   ( t )· N   3,P     bS   ( t )
 
     Following formulae were set;
 
 T   1,P     bS   ( t )=φ 1,P     bS   ( t )′/|φ 1,P     bS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   1 
                   , 
                   
                     P 
                     bS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           1 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                           ⁢             κ     1   ,     P   bS         ⁡     (   t   )       ·       N     1   ,     P   bS         ⁡     (   t   )         =       ⅆ     T     1   ,     P   bs             ⅆ   s         ;                       κ     1   ,     P   bS         ⁡     (   t   )       =       (       (       (       D     p   bS       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     1   ,     P   bS               ′     ⁢   3             ;                 a   1,P     bS   ( t )= S   1,P     bS     ″·T   1,P     bS   ( t )+κ 1,P     bS   ( t )· N   1,P     bS   ( t )
 
 T   2,P     bS   ( t )=φ 2,P     bS   ( t )′/|φ 2,P     bS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   2 
                   , 
                   
                     P 
                     bS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           2 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                           ⁢             κ     2   ,     P   bS         ⁡     (   t   )       ·       N     2   ,     P   bS         ⁡     (   t   )         =       ⅆ     T     2   ,     P   bS             ⅆ   s         ;                       κ     2   ,     P   bS         ⁡     (   t   )       =       (       (       (       D     P   bS       -       (       ∑     k   ,   l       ⁢       ɛ     rr       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     cc       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     2   ,     P   bS               ′     ⁢   3             ;                 a   2,P     bS   ( t )= S   2,P     bS     ″·T   2,P     bS   ( t )+κ 2,P     bS   ( t )· N   2,P     bS   ( t )
 
 T   3,P     bS   ( t )=φ 3,P     bS   ( t )′/|φ 3,P     bS   ( t )′|;
 
     
       
         
           
             
               
                 S 
                 
                   3 
                   , 
                   
                     P 
                     bS 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     t 
                     o 
                   
                   t 
                 
                 ⁢ 
                 
                   
                     
                       
                         φ 
                         
                           3 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ′ 
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
             ; 
           
         
       
     
                           ⁢             κ     3   ,     P   bS         ⁡     (   t   )       ·       N     3   ,     P   bS         ⁡     (   t   )         =       ⅆ     T     2   ,     P   bS             ⅆ   s         ;                       κ     3   ,     P   bS         ⁡     (   t   )       =       (       (       (       D     P   bS       -       (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )     ⁢     t   2         )     /     (       ∑     k   ,   l       ⁢       ɛ     ll       P   k     ,     P   l         ′     ⁢     ⅆ   t         )       )       1   2       )     -     0   /     S     3   ,     P   bS               ′     ⁢   3             ;                 a   3,P     bS   ( t )= S   3,P     bS     ·T   3,P     bS   ( t )+κ 3,P     bS   ( t )· N   3,P     bS   ( t )
 
(x 1 ,x 2 ,x 3 , t) is the coordinate system of the red blood cell in a neighbourhood O P     bS    of the basal Septum and δ(x 1 ,x 2 ,x 3 , t)=δ*(x 1 , t)·δ*(x 2 , t)·δ*(x 3 , t) where δ* is the dirac function and C 1,P     bS   , C 2,P     bS    and C 3,P     bS    are the graphs of φ 1,P     bS   (t), φ 2,P     bS   (t) and φ 3,P     bS   (t) respectively then the mechanical parameters of the red blood cells in the region O P     bS    are calculated by the following formulae:
 
     
       
         
           
             
               
                 
                   v 
                   
                     1 
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           1 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     1 
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       1 
                     
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           1 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   1 
                   , 
                   
                     P 
                     bS 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     1 
                   
                   , 
                   
                     P 
                     bS 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         1 
                         , 
                         
                           P 
                           bS 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     2 
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           2 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     2 
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       2 
                     
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           2 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   2 
                   , 
                   
                     P 
                     bS 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     2 
                   
                   , 
                   
                     P 
                     bS 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         2 
                         , 
                         
                           P 
                           bS 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   v 
                   
                     3 
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         T 
                         
                           3 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   n 
                   
                     3 
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       C 
                       3 
                     
                     , 
                     
                       P 
                       bS 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       
                         N 
                         
                           3 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ⊗ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 a 
                 
                   3 
                   , 
                   
                     P 
                     bS 
                   
                 
                 RBC 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     C 
                     3 
                   
                   , 
                   
                     P 
                     bS 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       a 
                       
                         3 
                         , 
                         
                           P 
                           bS 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⊗ 
                   
                     δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
     
     In an embodiment, the invention provides method for regionally making blood flow curve as described below; 
     For apical Anterior: 
     Setting v P     aA   (t)=(v 1,P     aA   (t), v 2,P     aA   (t), v 3,P     aA   (t)) as field velocity vectors of the blood in region O P     aA   , field of displacements in the real time at the same region is obtained by: 
     
       
         
           
             
               
                 r 
                 
                   P 
                   aA 
                 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           1 
                           , 
                           
                             P 
                             aA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           2 
                           , 
                           
                             P 
                             aA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           3 
                           , 
                           
                             P 
                             aA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
               ) 
             
           
         
       
     
     If, algebraic form of r P     aA    is called as BFC P     aA   ((x 1 ,x 2 ,x 3 , t)) then 
     
       
         
           
             
               X 
               
                 P 
                 aA 
               
             
             = 
             
               Spec 
               ⁡ 
               
                 ( 
                 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ] 
                     
                   
                   
                     
                       BFC 
                       
                         P 
                         aA 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     Similarly, 
     For mid Anterior 
     Setting v P     mA   (t)=(v 1,P     mA   (t), v 2,P     mA   (t), v 3,P     mA   (t)) as field velocity vectors of the blood in region O P     mA   , field of displacements in the real time at the same region is obtained by: 
     
       
         
           
             
               
                 r 
                 
                   P 
                   mA 
                 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           1 
                           , 
                           
                             P 
                             mA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           2 
                           , 
                           
                             P 
                             mA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           3 
                           , 
                           
                             P 
                             mA 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
               ) 
             
           
         
       
     
     If, algebraic form of r P     mA    is called as BFC P     mA    ((x 1 ,x 2 ,x 3 , t)) then 
     
       
         
           
             
               X 
               
                 P 
                 mA 
               
             
             = 
             
               Spec 
               ⁡ 
               
                 ( 
                 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ] 
                     
                   
                   
                     
                       BFC 
                       
                         P 
                         mA 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     For basal Anterior 
     Setting v P     bA   (t)=(v 1,P     bA   (t), v 2,P     bA   (t), v 3,P     bA   (t)) as field velocity vectors of the blood in region O P     bA   , field of displacements in the real time at the same region is obtained by: 
     
       
         
           
             
               
                 r 
                 
                   P 
                   
                     b 
                     ⁢ 
                     A 
                   
                 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           1 
                           , 
                           
                             P 
                             
                               b 
                               ⁢ 
                               A 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           2 
                           , 
                           
                             P 
                             
                               b 
                               ⁢ 
                               A 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           3 
                           , 
                           
                             P 
                             
                               b 
                               ⁢ 
                               A 
                             
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
               ) 
             
           
         
       
     
     If, algebraic form of r P     bA    is called as BFC P     bA   ((x 1 ,x 2 ,x 3 , t)) then 
     
       
         
           
             
               X 
               
                 P 
                 
                   b 
                   ⁢ 
                   A 
                 
               
             
             = 
             
               Spec 
               ⁡ 
               
                 ( 
                 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ] 
                     
                   
                   
                     
                       BFC 
                       
                         P 
                         
                           b 
                           ⁢ 
                           A 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     For the apical inferior 
     Setting v P     al   (t)=(v 1,P     al   (t), v 2,P     al   (t), v 3,P     al   (t)) as field velocity vectors of the blood in region O P     al   , field of displacements in the real time at the same region is obtained by: 
     
       
         
           
             
               
                 r 
                 
                   P 
                   al 
                 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           1 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           2 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           3 
                           , 
                           
                             P 
                             al 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
               ) 
             
           
         
       
     
     If, algebraic form of r P     al    is called as BFC P     al   ((x 1 ,x 2 ,x 3 , t)) then 
     
       
         
           
             
               X 
               
                 P 
                 aI 
               
             
             = 
             
               Spec 
               ⁡ 
               
                 ( 
                 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ] 
                     
                   
                   
                     
                       BFC 
                       
                         P 
                         aI 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     Similarly, 
     For mid Inferior 
     Setting v P     ml   (t)=(v 1,P     ml   (t), v 2,P     ml   (t), v 3,P     ml   (t)) as field velocity vectors of the blood in region O P     ml   , field of displacements in the real time at the same region is obtained by: 
     
       
         
           
             
               
                 r 
                 
                   P 
                   ml 
                 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           1 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           2 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           3 
                           , 
                           
                             P 
                             ml 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
               ) 
             
           
         
       
     
     If, algebraic form of r P     ml    is called as BFC P     ml   ((x 1 ,x 2 ,x 3 , t)) then 
     
       
         
           
             
               X 
               
                 P 
                 mI 
               
             
             = 
             
               Spec 
               ⁡ 
               
                 ( 
                 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ] 
                     
                   
                   
                     
                       BFC 
                       
                         P 
                         mI 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     For basal Inferior 
     Setting v P     bl   (t)=(v 1,P     bl   (t), v 2,P     bl   (t), v 3,P     bl   (t)) as field velocity vectors of the blood in region O P     bl   , field of displacements in the real time at the same region is obtained by: 
     
       
         
           
             
               
                 r 
                 
                   P 
                   bl 
                 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           1 
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           2 
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           3 
                           , 
                           
                             P 
                             bl 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
               ) 
             
           
         
       
     
     If, algebraic form of r P     bl    is called as BFC P     bl   ((x 1 ,x 2 ,x 3 , t)) then 
     
       
         
           
             
               X 
               
                 P 
                 bI 
               
             
             = 
             
               Spec 
               ⁡ 
               
                 ( 
                 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ] 
                     
                   
                   
                     
                       BFC 
                       
                         P 
                         bI 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     For the apical Lateral 
     Setting v P     aL   (t)=(v 1,P     aL   (t), v 2,P     aL   (t), v 3,P     aL   (t)) as field velocity vectors of the blood in region O P     aL   , field of displacements in the real time at the same region is obtained by: 
     
       
         
           
             
               
                 r 
                 
                   P 
                   aL 
                 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           1 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           2 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           3 
                           , 
                           
                             P 
                             aL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
               ) 
             
           
         
       
     
     If, algebraic form of r P     aL    is called as BFC P     aL   ((x 1 ,x 2 ,x 3 , t)) then 
     
       
         
           
             
               X 
               
                 P 
                 aL 
               
             
             = 
             
               Spec 
               ⁡ 
               
                 ( 
                 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ] 
                     
                   
                   
                     
                       BFC 
                       
                         P 
                         aL 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     Similarly, 
     For mid Lateral 
     Setting v P     mL   (t)=(v 1,P     mL   (t), v 2,P     mL   (t), v 3,P     mL   (t)) as field velocity vectors of the blood in region O P     mL   , field of displacements in the real time at the same region is obtained by: 
     
       
         
           
             
               
                 r 
                 
                   P 
                   mL 
                 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           1 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           2 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           3 
                           , 
                           
                             P 
                             mL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
               ) 
             
           
         
       
     
     If, algebraic form of r P     mL    is called as BFC P     mL   ((x 1 ,x 2 ,x 3 , t)) then 
     
       
         
           
             
               X 
               
                 P 
                 mL 
               
             
             = 
             
               Spec 
               ⁡ 
               
                 ( 
                 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ] 
                     
                   
                   
                     
                       BFC 
                       
                         P 
                         mL 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     For basal Lateral 
     Setting v P     bL   (t)=(v 1,P     bL   (t), v 2,P     bL   (t), v 3,P     bL   (t)) as field velocity vectors of the blood in region O P     bL   , field of displacements in the real time at the same region is obtained by: 
     
       
         
           
             
               
                 r 
                 
                   P 
                   bL 
                 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           1 
                           , 
                           
                             P 
                             bL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           2 
                           , 
                           
                             P 
                             bL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           3 
                           , 
                           
                             P 
                             bL 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
               ) 
             
           
         
       
     
     If, algebraic form of r P     bL    is called as BFC P     bL   ((x 1 ,x 2 ,x 3 , t)) then 
     
       
         
           
             
               X 
               
                 P 
                 bL 
               
             
             = 
             
               Spec 
               ⁡ 
               
                 ( 
                 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ] 
                     
                   
                   
                     
                       BFC 
                       
                         P 
                         bL 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     For the apical Septum 
     Setting v P     aS   (t)=(v 1,P     aS   (t), v 2,P     aS   (t), v 3,P     aS   (t)) as field velocity vectors of the blood in region O P     aS   , field of displacements in the real time at the same region is obtained by: 
     
       
         
           
             
               
                 r 
                 
                   P 
                   aS 
                 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           1 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           2 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           3 
                           , 
                           
                             P 
                             aS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
               ) 
             
           
         
       
     
     If, algebraic form of r P     aS    is called as BFC P     aS   ((x 1 ,x 2 ,x 3 , t)) then 
     
       
         
           
             
               X 
               
                 P 
                 aS 
               
             
             = 
             
               Spec 
               ⁡ 
               
                 ( 
                 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ] 
                     
                   
                   
                     
                       BFC 
                       
                         P 
                         aS 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     Similarly, 
     For mid Septum 
     Setting v P     mS   (t)=(v 1,P     mS   (t), v 2,P     mS   (t), v 3,P     mS   (t)) as field velocity vectors of the blood in region O P     mS   , field of displacements in the real time at the same region is obtained by: 
     
       
         
           
             
               
                 r 
                 
                   P 
                   mS 
                 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           1 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           2 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           3 
                           , 
                           
                             P 
                             mS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
               ) 
             
           
         
       
     
     If, algebraic form of r P     mS    is called as BFC P     mS   ((x 1 ,x 2 ,x 3 , t)) then 
     
       
         
           
             
               X 
               
                 P 
                 mS 
               
             
             = 
             
               Spec 
               ⁡ 
               
                 ( 
                 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ] 
                     
                   
                   
                     
                       BFC 
                       
                         P 
                         mS 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     For basal Septum 
     Setting v P     bS   (t)=(v 1,P     bS   (t), v 2,P     bS   (t), v 3,P     bS   (t)) as field velocity vectors of the blood in region O P     bS   , field of displacements in the real time at the same region is obtained by: 
     
       
         
           
             
               
                 r 
                 
                   P 
                   bS 
                 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               ( 
               
                 t 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           1 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           2 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
                 , 
                 
                   
                     ∫ 
                     
                       t 
                       o 
                     
                     s 
                   
                   ⁢ 
                   
                     
                       
                         υ 
                         
                           3 
                           , 
                           
                             P 
                             bS 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
               ) 
             
           
         
       
     
     If, algebraic form of r P     bS    is called as BFC P     bS   ((x 1 ,x 2 ,x 3 , t)) then 
     
       
         
           
             
               X 
               
                 P 
                 bS 
               
             
             = 
             
               Spec 
               ⁡ 
               
                 ( 
                 
                   
                     R 
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         
                           x 
                           3 
                         
                         , 
                         t 
                       
                       ] 
                     
                   
                   
                     
                       BFC 
                       
                         P 
                         bS 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             x 
                             3 
                           
                           , 
                           t 
                         
                         ) 
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     The scheme of blood flow curve is as below
 
 X   Blood flow in LV   =X   P     aA     ∪X   P     mA     ∪X   P     bA     ∪X   P     al     ∪X   P     ml     ∪X   P     bl     ∪X   P     aL     ∪X   P     mL     ∪X   P     bL      ∪X   P     aS     ∪X   P     mS     ∪X   P     bS   .