**3. Results**

### *3.1. Computation of C*0 *and* Γ0

Eighteen aortic conditions were studied in 14 patients, of which 4 had normal aortas, 6 had lesions located in the proximal regions, and 4 had lesions localized in the distal regions. Three-dimensional reconstructions of the aortas were obtained using MSCT.

Each reconstruction is a surface in STL format. The surfaces presented in this format are a set of conjugated triangles with indication of the normal to them. The following algorithm was used to calculate the *C*0 and Γ0 values for these reconstructions:

1. The initial matrix with the coordinates of the triangles that make up the surface of the aortic duct was presented as a matrix A with dimensions [num\_rows, num\_features]. Here num\_rows is the total number of triangles that make up the surface, and num\_features is the number of points representing each such triangle (num\_features = 9 for 3D space).

2. The computed matrix A was projected onto a plane by the PCA method (there was a decrease in the dimension from 9 to 2). Next, the length of the 2d-projection of the aortic flow channel was measured, and on the distance from the beginning of the aorta by 10% and 90% of the entire length of the aorta, the aortic radius was measured. Thus, two pairs of values (*<sup>z</sup>*1,*r*1) and (*<sup>z</sup>*2,*r*2) were obtained, which are important for *C*0 and Γ0 computation according to expressions (5) and (6).

Normally, for patients without obvious aortic pathology, the streamlines can be plotted as follows (Figure 1a,b).

**Figure 1.** *Cont*.

**Figure 1.** (**a**) Streamlines of the swirling flow in the normal aorta in longitudinal-radial projection. (**b**) Streamlines of the swirling flow in the normal aorta in the axial-radial projection.

Figure 1b shows a plot of *C*0 and Γ0 values depending on the location of aortic pathology. Table 1 shows the corresponding values of the parameters *C*0 and Γ0.


**Table 1.** Values of constants *C*0 and Γ0 for the studied aortas, 'loc'—localization of aortic lesion (down—distal aneurysm, up—proximal aneurysm, norm—no pathology).

As can be seen in Figure 2, a pair of *C*0 and Γ0 values can serve as a quantitative criterion for identifying aortic pathology. All three considered cases (pathology in the descending aorta, pathology in the ascending aorta and the norm) are linearly separable.

**Figure 2.** Comparison of *C*0 and Γ0 values for aortas from Table 1. Blue triangles indicate aortas with pathology in the proximal sections, orange triangles—aortas with pathology in the distal sections, and green circles—aortas without severe pathology.

In the case of a pathological disturbance of the vascular bed in the descending section, the value of the circulation Γ0 of the swirling flow increases significantly. At the same time, a *C*0 raising can be observed followed by the increase in the transverse gradients of the blood flow velocity. Fluctuations of *C*0 and Γ0 values may reflect the action of compensatory and regulatory mechanisms of the cardiovascular system. However, the actions of these mechanisms are inevitably associated with excessive energy consumption to maintain the flow structure and can also lead to an increased force impact on the aortic wall.

In the case of pathology in the ascending region, one can observe a slight increase in the Γ0 value and a relatively small (compared with the pathology of the descending region) increase in *C*0 value.

However, the parameters *C*0 and Γ0 do not unequivocally allow the establishment of the fact of pathological remodeling of the aortic duct. In Figure 1a,b, the dots representing aortas with distal damage lie very close to the dots corresponding to the normal aorta.

#### *3.2. Approximation of the Aorta Flow Channel by a Parametric Spiral*

Figure 3a,b show the result of plotting the central line for an aorta with no obvious pathological disorders and an aorta with pathology, respectively.

**Figure 3.** (**a**) Three-dimensional STL—reconstruction of the flow channel of the aorta without severe pathology. A central line is built inside the flow channel. (**b**) Three-dimensional STL—reconstruction of the flow channel of the aorta with severe pathology of the distal sections. A central line is built inside the flow channel.

Each center line is a matrix with dimensions [num\_points, 3], where num\_points is the number of points in the center line, and *3* is the number of spatial dimensions in the Cartesian coordinate system. The resulting lines must be approximated by some spiral curves to obtain the characteristic parameters. The search for spirals was carried out in the class of hyperbolic spirals described by the following relation in the polar coordinate system:

$$r = \left(\frac{a}{\varphi}\right)^{power} + bias\tag{11}$$

In relation (11), the unknown parameters are (*a*, *power*, *bias*).

The hyperbolic class of spirals was chosen since the streamlines of the swirling blood flow in the aorta in the radial-axial projection are described by a hyperbolic spiral, and the similarity principle indicates that the flow channel, in which the flow evolves without the formation of separation and stagnant zones, must have similar geometry.

For each central line, an approximating spiral was constructed in accordance with the following algorithm:


$$r = 
\sqrt{x^2 + y^2} 
\cdot 
\begin{array}{c} 
\sqrt{x^2 + y^2} 
\end{array}$$

Here *atan*<sup>2</sup>(*y*, *x*)—2-argument arctangent used to translate Cartesian coordinates into polar coordinates. This arctangent can be stated as follows:

$$\operatorname{atan2}(y,\,\,\mathbf{x}) = \begin{cases} 2\arctan\left(\frac{y}{\sqrt{\mathbf{x}^2 + y^2} + \mathbf{x}}\right), \,\,\, if\,\,\mathbf{x} > 0 \,\, and \,\, y \neq 0, \\\ \pi, \,\, if\,\,\mathbf{x} < 0 \,\, and \,\, y = 0, \\\ \text{undefined}, \,\,\, if\,\,\mathbf{x} = 0 \,\, and \,\, y = 0 \end{cases}$$

Using the least squares method, the parameters (*a*, *power*, *bias*) from expression (11) were selected in such a way that the polar representation of the centerline\_proj line is most accurately described by the parametric hyperbolic spiral (11).

The obtained parameters of the approximating spiral (*a*, *power*, *bias*), the coefficient of determination *R*2, and the standard deviation (mae) of the real line centerline\_proj from the approximating spiral were entered in the resulting table.

Based on the calculated five parameters *a*, *power*, *bias*, *R*2, *mae* . 2 synthetic parameters were calculated by the PCA method (*feature\_1, feature\_2*). These parameters store all the necessary information about the quantitative differences in the parameters *a*, *power*, *bias*, *R*2, *mae* for the normal aorta and in the presence of a pathological change in the vascular bed and allow visual interpretation of these differences.

The application of the formulated algorithm for one central line looks like this:

The plotting of the initial three-dimensional central line centerline (white line) is performed in Figure 4.

**Figure 4.** STL-reconstruction of the aortic flow channel for a patient without severe pathology and the central line inside the channel.

The graph of the line centerline\_proj was plotted, with the projection of the central line onto the plane in Cartesian and polar coordinate systems (Figure 5).

**Figure 5.** On the left—the projection of the central line in the Cartesian coordinate system, on the right—in the polar coordinate system.

Using the least squares method, the parameters of the approximating spiral were calculated. In Figure 6, the original curve and its approximation are plotted in the polar coordinate system.

**Figure 6.** Comparison of a flat projection of the central line of the aorta and its approximation by a spiral in polar coordinates. The orange line is the projection of the central line, the blue line is the fitting curve.

Approximating spirals for all central lines were constructed using an identical algorithm. The results obtained are shown in Table 2.

**Table 2.** Comparison of the geometric characteristics of the approximation of the central line of the aorta.


(a, power, bias)—parameters of the approximating hyperbolic spiral for the projection of the central line from expression (11), mae—the value of the standard deviation of the approximating curve from the projection of the central line, R2—coefficient of determination for a specific approximation, feature\_1, feature\_2—derived parameters, obtained from a, power, bias, R2, mae by PCA method.

As can be seen from the table, the value of the coefficient of determination *R*<sup>2</sup> for aortas without noticeable remodeling is higher, and the value of the standard deviation of the approximation *mae* is lower than for aortas with pathological disorders of the vascular bed. For normal aortas, the coefficient of determination exceeds 0.95, which indicates high approximation accuracy. For aortas with severe pathology of the vascular bed, the coefficient of determination exceeds 0.77, which indicates the significance of the chosen approximation method.

In Figure 6 was depicted approximation for normal aorta denoted as mir\_s in Table 2. Other approximations are depicted on Figures 7–22 The notations and lines color are the same, as on Figure 6.

**Figure 7.** Approximation for are\_s.

**Figure 8.** Approximations for ino\_s.

**Figure 9.** Approximation for ino\_d.

**Figure 10.** Approximations for she\_a.

**Figure 11.** Approximation for are\_d.

**Figure 12.** Approximations for bor\_a.

**Figure 13.** Approximation for mal\_a.

**Figure 14.** Approximations for pav\_a.

**Figure 15.** Approximation for bar\_a.

**Figure 16.** Approximations for zag\_a.

**Figure 17.** Approximation for mir\_d.

**Figure 18.** Approximations for gor\_d.

**Figure 19.** Approximation for poz\_a.

**Figure 20.** Approximations for hom\_a.

**Figure 21.** Approximation for lar\_s.

**Figure 22.** Approximations for lar\_d.

The values of the derived quantitative features feature\_1 and feature\_2 make it possible to unambiguously separate aortas without a noticeable pathological disorder and aortas with a violation of the geometry of the vascular bed (Figure 23).

The values of quantitative features (*feature\_1, feature\_2*), which were calculated by approximating the central line of the aortic flow channel with a hyperbolic spiral, make it possible to clearly separate the aorta without pronounced pathological remodeling and the aorta with pathological disturbance of the vascular bed. However, the obtained parameters do not allow one to reliably divide aortas according to the type of pathological remodeling (lesion in the proximal or distal sections).

As one can see, there is one obvious outlier on a plot from Figure 23. It is caused by a severely damaged aortic duct in the distal regions. As a result, the proposed algorithm can't properly handle such altered geometry and issues biased values for features.

**Figure 23.** Comparison of quantitative characteristics of geometry for aortas. Blue triangles indicate aortas with pathology in the proximal sections, green triangles—aortas with pathology in the distal sections, and orange circles—aortas without severe pathology.

With a pathological violation of the geometry of the flow channel of the aorta, an area is formed in the local section of the channel, the radius of which significantly exceeds the radius of the same section in the norm. As follows from relations (2.1), for the volume of the swirling flow that fills this additional "pathological" region, at constant values of *C*0 and Γ0 the radial and azimuthal velocity components will be higher than in the normal case. An increase in the total linear velocity vector in the presence of severe pathological remodeling of the aortic duct was experimentally confirmed. However, if we assume that the parameters *C*0 and Γ0 for a pathological case normally coincide, the increase in the total vector of blood flow velocity in pathology will exceed the experimentally observed changes. Therefore, the value of Γ0 in case of a pathological violation of the geometry of the flow channel will exceed the normal values to compensate for the increase in the radial size of the area with the aneurysm. The value of *C*0 with pathology will increase slightly. This will lead to an insignificant increase in the radial and longitudinal departure velocity, which will cause an increase in the energy spent to maintain the evolution of the swirling flow. However, at the same time, an increase in the value of *C*0 leads to a decrease in the viscous radius of the swirling flow (the region in which the influence of viscosity is strong). This will inevitably lead to a decrease in the energy spent on maintaining the twist. As a result, in case of a pathological violation of the geometry of the flow channel, we will ge<sup>t</sup> a small increase in the energy spent on maintaining the swirling blood flow; however, this value lies within the limits that approximately correspond to indirect observations.
