## 2 Materials and Methods
If these vectors are approximated by a dependence of the form:  
\[(z_{0}+z_{i})*r_{i}^{p}=c, \tag{1}\]  
the following optimum values were obtained:  
\[z_{0}=0.0108\;m,\;B=0.0001\;m^{3},\;p=1.944, \tag{2}\]  
which were selected within the following limits: [(-0.05, 1 \(\times\) 10\({}^{-5}\), 0.0), (0.05, 1 \(\times\) 10\({}^{-3}\), 3.0)] in accordance with previously obtained experimental data [12]. The value of \(z_{0}\) in this case is 10 mm, which corresponds to the distance from the point of formation of the twisted jet to the beginning of the cylindrical coordinate system in which this jet can be described, and the value of the constant is proportional to the volume of the jet in the segment from the left atrium to the ventricle, as if this jet were formed in the volume of the entire hyperboloid and not in its segmental part. Therefore, the value of the constant in this segment is greater than in the left ventricle-aorta segment (see below). When studying the shape of the left ventricular and aortic system with the aortic valve open, we measured spheres inscribed successively in the lumen of the channel, because the distortions of the spatial axis of the system in the area of the ventricle and the aortic valve do not allow us to determine the position of the normal plane of the vessel cross-section. The essence of the approach used is to inscribe in 3D object projections the connected projections of a number of spheres. In other words, the study was carried out using methods of descriptive geometry. The scheme shown in Figure 2b shows the construction of projections of the sphere \(\alpha(A)\) touching the surface at point \(A\) on two connected projections of the aorta. The projections \(\alpha(A)_{2}\) and \(\alpha(A)_{3}\) have been constructed taking into account the projective relationship of the images obtained. The circle \(t_{2}\) is used to find the normal to the outline of the surface. The center of the sphere must lie on this normal, at the same distance from the two opposite outlines, and the sphere outline must contain the point A\({}_{2}\). In order to study the evolution of the blood jet traveling from the left ventricle to the aorta, the construction of fourteen spheres consecutively inscribed in the system of the left ventricle and the aorta was carried out (Figure 3). Figure 3b also shows the circumferential formations of the desired surface as an approximate surface \(\gamma\). Also in Figure 3b, the \(i\)-axis of the left ventricle-aorta system is shown in two connected projections, which are sufficient to construct its three-dimensional model. Least-squares analyses of the data presented in Figure 3 also allowed the following optimal values of the constants of \(zt^{p}=B\) to be determined:  
\[z_{0}=0.047m,\;B=0.00007m^{3},\;p=2.09 \tag{3}\]  
The values of \(B\) and \(z_{0}\) correspond to those given in the article [12]. The study of the shape of the aortic flow channel with the aortic valve closed was carried out in the same way. The spheres were inscribed in the aortic lumen and then the  
Figure 3: The result of constructing the spheres inscribed in the left ventricle and aorta. (**a**) three projections of the spheres combined with the image of the ventricle and aorta, the projections of the spheres are shown by the red lines; (**b**) two projections of the spheres and the broken ventricle-aorta axis, shown by the blue dashed line; (**c**) a graph of the dependence of the radius (\(r\)) of the sphere from the \(\{\alpha\}\) family on the distance along the curve (\(l_{\rm i}=z_{\rm i}\)). Figure 2: Left heart and aortic sections with open aortic valve: (**a**) two orthogonal projections of the aorta and left heart sections; (**b**) construction of two projections of the sphere \(\alpha(A)\) inscribed in the image of the aortic section. An arbitrary point \(A\) lying on the contour \(u\) of the original model is taken (\(A\in u\)). A circle \(t\) with center at \(A\) and arbitrary radius is constructed, and its intersection points with \(u\) are found (\(C=t\cap u\), \(D=t\cap u\)).