## 3 Results  
The values (2), (3), (4) of the degree (\(p\)) of the variable (\(r\)) obtained in the experiments indicate the uniformity of the shaping for the studied surfaces at the moment of maximum productivity of the blood flow. All the values obtained for the degree of \(r\) are close to two. Consequently, it is reasonable to use a degree \(p\) equal to two when constructing the approximating surface. Equation (1) can be written as a function of \(r\) on the argument \(z\). Then Equation (1) takes the form:  
\[\mathbf{r}=\left(\frac{\mathbf{c}}{z_{0}+z}\right)^{\frac{1}{2}}. \tag{5}\]  
It should be noted that the function under consideration only makes sense for positive values of function and argument; therefore, the loss of roots in the equation does not affect the final result when going from (1) to (5). The value of the constant B was corrected because the value of the degree (\(p\)) was rounded to two. As a result, we obtained the following equations for calculating the radius of the atrial and ventricular-aortic flow channels, respectively:  
\[\mathbf{r}=\left(\frac{\mathbf{0.000010}}{-\mathbf{0.0032}+z}\right)^{\frac{1}{2}},\mathbf{r} =\left(\frac{\mathbf{0.000024}}{\mathbf{0.0051}+z}\right)^{\frac{1}{2}}, \tag{6}\]  
where \(z\) is the distance along the spatial curve of the vessel axis. Equation (6) was used to calculate the data shown in Figure 4.  
Based on the data obtained (Figure 4), it can be concluded that Equation (6) can be used to construct target approximation surfaces. For a more visual representation of the direction of blood flow from the pulmonary veins, they are modeled conventionally (Figure 5). The images obtained show the relationship between the direction of flow and the shape of the flow channel, confirming the effectiveness of the hydrodynamic model chosen [5]. The segment of the surface approximating the atrial shape (Figure 5) can be represented as an image of a straight confuser [13] in accordance with 'oblique symmetry' [14]. The atrial current lines are inscribed based on the position of the pulmonary vein axes and the axis of the hyperboloid, a segment of which is the left atrial channel. This construction is empirical. The analytical solution should be based on the instantaneous values of the functions of the current lines of the twisted jet in the left atrium. In order to construct a surface that approximates the shape of the ventricle-aorta system, it is necessary to obtain a model of its spatial axis curve. In this study, a spline curve constructed on the centers of the spheres shown in Figure 3 was used for this purpose. The choice of spline construction method [15; 16; 17] remains open in this case. In our solution, the rational Bezier curve is used. Circular sections of the surface were defined in normal planes and the surface was then drawn on them using the appropriate tool in the computer-aided design system. Equation (6) and the data presented in Figure 4 were used to determine the radii of the forming circles. Figure 4: Correspondence between the graphs of the approximation curves (red) and the curves constructed on the basis of their measurements (blue): (**a**) change in the radius of curvature of the streamlined surface of the left atrium along the axis of the swirling jet; (**b**) change in the radius of the aortic left ventricular system along the axis of the flow channel. Figure 5: Joint images from four angles: left atrium, segment of the analytical approximation surface (in purple) and conventional lines of blood flow (in red) from the pulmonary veins to the left ventricle at the moment of mitral valve opening. The resulting surface fragment models approximating the aorta, left ventricle, and left atrium are superimposed on the original STL model in Figure 6. In the figure, the left atrial cavity (pink) is delineated as a segment of a hyperboloid, the flow portion of the left ventricular and aortic cavities is obtained by sphere inscription (beige), and the blue surface corresponds to the concave streamlined surface lying at the jet initiation site.