Obtaining Vortex Formation in Blood Flow by Particle Tracking: Echo-PV Methods and Computer Simulation

Abstract

Micrometer-sized particles are widely introduced as fluid flow markers in experimental studies of convective flows. The tracks of such particles demonstrate a high contrast in the optical range and well illustrate the direction of fluid flow at local vortices. This study addresses the theoretical justification on the use of large particles for obtaining vortex phenomena and its characterization in stenotic arteries by the Echo Particle Velocimetry method. Calcite particles with an average diameter of 0.15 mm were chosen as a marker of streamlines using a medical ultrasound device. The Euler–Euler model of particle motion was applied to simulate the mechanical behavior of calcite particles and 20 µm aluminum particles. The accuracy of flow measurement at vortex regions was evaluated by computational fluid dynamics methods. The simulation results of vortex zone formation obtained by Azuma and Fukushima (1976) for aluminum particles with the use of the optical velocimetry method and calcite particles were compared. An error in determining the size of the vortex zone behind of stenosis does not exceed 5%. We concluded that the application of large-size particles for the needs of in vitro studies of local hemodynamics is possible.

1. Introduction

The high mortality rate in the world from cardiovascular diseases has lead to numerous studies of hemodynamic flows in coronary vessels. Distortions of the blood vessel lumen, including physiologic bifurcations and bends, can cause localized blood flow disturbances. The abnormal hemodynamic forces associated with these disturbances may play an important role in the pathogenesis of focal vascular lesions such as atherosclerosis, extrinsic thrombus, poststenotic dilatation, and aneurysm [1,2,3,4,5]. In particular, a number of studies have demonstrated the appearance of flow separation, vortices and turbulence, and flow recirculation in the post-stenotic region [6,7,8,9]. These findings are of great interest for studying the contribution of hemodynamic factors to the formation of stenosis.

In recent decades, the development of CFD (Computational Fluids Dynamic) methods has resulted in an increase in the number of works on the modeling of local hemodynamics, including ones with the use of patient-specific data [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. To account for the rheology of blood and the interaction of liquid and solid with the walls of blood vessels, many complex multiphase and multiparametric models have been developed [26,27,28,29]. These models, based mainly on the numerical solution of the Navier–Stokes equations, allow the complete detailed reconstruction of the local hemodynamics in the area of stenosis.

One of the most interesting features of the local hemodynamics in damaged arteries is the formation of a low-velocity vortex behind the stenosis. Due to the non-Newtonian nature of blood, the vortex zone is associated with a local increase in viscosity and lower near-wall shear stresses [9,15,30,31,32,33,34,35]. The complex blood rheological effects result in the appearance of a cell-free layer (CFL) [19,36] with a lower viscosity compared to the core region. The thickness of CFL significantly affects the velocity profiles and dimensions of the vortex zone. The quantitative characterization of the vortex zone is important for the development of stenosis theory, diagnostics, and clinical applications. As the nature of the CFL is still not clear, the size of the vortex region is difficult to predict without in silico simulations.

In general, a reliable investigation on the effect of local hemodynamics in vessels with stenosis on the development of disease and its treatment methods requires an integrated approach combining in vivo, in vitro, and in silico studies (Figure 1).

Since the complex models of local hemodynamics usually contain a number of empirical coefficients, in vitro experiments of model verification are necessary. For this purpose, optical particle velocimetry (PV) [37,38,39,40,41,42] and ultrasonic PV (Echo-PV) [43,44] technologies are widely used. The optical PV method operates with a fine contrast impurity (polystyrene, polyamide, or glass beads with size 0.2–100 µm). Such methods are highly accurate and informative for assessing local hemodynamics. The main disadvantage of optical methods is the requirement for optical transparency of the vessel wall and fluid. In contrast, the Echo-PV method allows measurements in optically opaque media, making it most suitable for flow studies in phantom vessels or in vivo.

At the same time, Echo-PV requires particles with a high reflectivity of ultrasonic waves. In particular, polystyrene beads of approximately 100 µm in diameter or air microbubbles of the same size are suitable for in vitro or in vivo studies, respectively [45]. Meanwhile, the accuracy of the optical PV method for assessing the peculiarities of vortex zone behind the stenosis has been confirmed experimentally [1], while the suitability of the Echo-PV method for this purpose is not yet determined. Therefore, the last stage in the methodological chain presented in Figure 1 is necessary to evaluate the accuracy of vortex zone formalization by Echo-PV using large echo contrast particles as stream markers.

This study addresses the theoretical justification on the use of large particles for obtaining vortex phenomena and its characterization in stenotic arteries by the Echo-PV method. First, we assessed the echogenicity of calcite particles with an average diameter of approximately 0.15 mm as a marker of streamlines. We show that such particles are well visualized by a medical ultrasound device and can be used to detect the vortex zone behind the stenosis. Then, we applied the Euler–Euler model of particle motion to simulate the mechanical behavior of calcite particles at the experimental conditions and compared the results of our simulations with the data of a classic in vitro study by Azuma and Fukushima [1] obtained with aluminum microparticles of 20 µm in diameter and their optical imaging. Finally, we show that large particles provide an error in the measured dimensions of the vortex zone within 5%.

2. Materials and Methods

2.1. Experimental Setup for Particle Echogenicity Testing in a Fluid Flow

To evaluate the ability of particle visualization in fluid flow through vessels of complex geometry by ultrasonography, an experimental setup of laboratory design was used. A piece of silicon tube with an inner diameter of 6 mm was inserted into a silicone tube with a higher inner diameter (8 mm) to simulate arterial stenosis. The silicon tube was connected to a reservoir filled with a suspension containing distilled water and calcite particles with a volume fraction of 0.1%. The calcite particles KoelgaCarb KM-200 (JSC «Koelgamramor», Chelyabinsk, Russia) were close to spherical shape, with a diameter 0.1–0.2 mm. The fluid flow rate of 500 mL/s was set by the height of the water column. The constant level of the water column in the reservoir was provided by the continuous addition of suspension into the reservoir from a buffer tank with the help of a peristaltic pump. For details of the hydrodynamic system, see [46].

A fragment of the silicon tube with a model stenosis was immersed in a polystyrene cuvette filled with 500 mL of water. A SIEMENS 7.5L45s Prima/Adara linear transducer of an ultrasonic medical device Sonoline Adara (Siemens, Munich, Germany) was immersed in the same cuvette at a distance of 40 mm from the silicone tube. The following hardware settings of the ultrasonic apparatus were used: dynamic range—66 dB, operating frequency—10 MHz, wavelength—0.15 mm, gain—19 dB, and power—50%. Two-dimensional images of the samples were recorded by video for several seconds at a rate of 25 frames per second and resolution of 720 × 480 pixels. The brightness was measured in gray scale in a range from 30 to 255 arbitrary units with the use of specially designed software. For details of the echolocation, see [47]. Calcite particles were captured and showed good echogenicity, so it was possible to capture the tracks of individual particles (Figure 2).

2.2. Liquid–Particle Model: Equations Describing the Processes in the Dispersed and Liquid Phases

A model for the propagation of marker particles in fluid flow was developed using the the Euler–Euler concept. Due to the small impurity concentration, the influence of particles on the fluid flows can be neglected. The model takes into account the change of hydrodynamic force on the particles from the liquid side as its diameter increases through coefficient $C_D$ (Equations (3) and (4)) in the drag force.

The advection equation for the dispersed phase can be written as

$$\frac{\partial n}{\partial t}+\nabla ·\left({\mathit{V}}_{d}n\right)=0,$$

(1)

where n$\left[{\mathrm{m}}^{-3}\right]$ is the concentration of the dispersed phase (concentration of aluminum particles and calcite particles) and ${\mathit{V}}_d$$\left[\mathrm{m}{\mathrm{s}}^{-1}\right]$ is the velocity of its motion.

The momentum transfer of the dispersed phase (aluminum particles, calcite particles) is expressed by the non-homogeneous convection equations for the conservative variable ${\mathit{V}}_{d}n$:

$$\begin{array}{cc}\hfill & m_d\frac{\partial \left({\mathit{V}}_{d}n\right)}{\partial t}+m_d\left(n{\mathit{V}}_d·\nabla \right){\mathit{V}}_d=-n\frac{\pi d^3}{6}\nabla P+{\mathit{F}}_D,\hfill \\ \hfill & {\mathit{F}}_D=n{\rho }_c{C}_D\frac{\pi d^2}{8}\left|{\mathit{V}}_c-{\mathit{V}}_d\right|\left({\mathit{V}}_c-{\mathit{V}}_d\right).\hfill \end{array}$$

(2)

In Equation (2), $m_d$$\left[\mathrm{kg}\right]$ is the local mass of particles, d$\left[\mathrm{m}\right]$ is the particle diameter, P [Pa] is the fluid pressure measured relative to the hydrostatic pressure, ${\rho }_c$$\left[\mathrm{kg}{\mathrm{m}}^{-3}\right]$ is the density of the continuous phase, ${\mathit{F}}_D$$\left[\mathrm{N}{\mathrm{m}}^{-3}\right]$ is the drag force, $C_D$ is the drag coefficient, and $V_c$$\left[\mathrm{m}{\mathrm{s}}^{-1}\right]$ is the velocity of continuous phase.

The movement of dispersed phase particles is primarily determined by the drag force, $F_D$, which prevents the movement. The drag coefficient in the equation for $F_D$ shows the relationship between the drag force and the inertial forces of the supporting medium. $C_D$ depends on many parameters, but it has been shown that, for particles of a particular shape, it is a function of only the Reynolds number ($Re$) [48]. The Reynolds number is the ratio of the inertial forces to the internal friction forces arising from the particle motion. In the range of small values of $Re$, the inertial forces make a negligibly small contribution to the drag force, which is completely determined by the viscosity of the fluid.

It has been found that the $C_D$ coefficient is approximately constant when the inertial forces are much larger than the internal friction forces (large values of $Re$). However, as can be seen from Figure 3, $C_D$ increases significantly as $Re$ decreases.

Figure 3 is the result of numerous experiments with sphere-shaped particles and can be divided into three regions: (1) $Re>500$, where $C_D\approx 0.44$ and approximately constant (Newton’s law is fulfilled); (2) intermediate region $(0.2<Re<500)$, where $C_d$ depends significantly on $Re$; (3) $Re<0.2$, where $C_D=24/Re$ (Stokes law).

In the range of small values of $Re$, the inertial forces make a negligibly small contribution to the drag force, which is completely determined by the viscosity of the fluid.

The applicability of Stokes law is limited by the velocity of particles in fluid: the motion must be so slow that the inertial forces are small compared to the drag forces. Large particles are capable of accelerated motion. The rapid movement of particles can lead to the development of local micro-turbulence, in which the Stokes law is no longer observed.

In this study, for each dispersed phase (aluminum particles, calcite particles), $Re$ was calculated in order to select a model for the drag coefficient based on the calculations. For aluminum particles with a diameter of 20 µm, $Re=0.19$, for calcium particles with a diameter of 200 µm, $Re=1.9$. From the results, it can be seen that, for aluminum particles, Stokes law is satisfied and the drag coefficient can be calculated as Equation (3). For larger calcite particles, the Stokes law is not fulfilled and the resistance coefficient, $C_D$, depends significantly on $Re$ in the intermediate region (see Figure 3). For calcite particles, the drag coefficient was calculated using the empirical Formula (4) obtained by Klyachko [50] for the range $0.2<Re<500$.

For the drag coefficient of the aluminum particle cloud, the following expression [51] is used:

$$\begin{array}{cc}\hfill & C_D=\frac{24}{R{e}_d}+0.44.\hfill \end{array}$$

(3)

The dependence of $C_D\left(Re\right)$ for the calcite particle cloud was described using the empirical Klyachko formula [50]:

$$\begin{array}{cc}\hfill & C_D=\frac{24}{R{e}_d}+\frac{4}{R{e}_d^{1/3}}.\hfill \end{array}$$

(4)

In Equations (3) and (4), $Re$ is defined as

$$Re=\frac{{\rho }_c\left|{\mathit{V}}_c-{\mathit{V}}_d\right|d}{{\mu }_c},$$

(5)

where ${\mu }_c$ is the molecular viscosity dynamic coefficient of the continuous phase, $\left[\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}\right]$.

For liquid fraction, the continuity equation can be written as

$$\frac{\partial \left({\phi }_c{\rho }_c\right)}{\partial t}+\nabla ·\left({\phi }_c{\rho }_c{\mathit{V}}_c\right)=0,$$

(6)

where ${\phi }_c$ is the relative volume of the liquid phase. The momentum equation takes the form of:

$$\begin{array}{cc}\hfill & \frac{\partial \left({\phi }_c{\rho }_c{\mathit{V}}_c\right)}{\partial t}+\left({\phi }_c{\rho }_c({\mathit{V}}_c·\nabla ){\mathit{V}}_c\right)=-{\phi }_c\nabla p+\nabla ·\left({\phi }_c\mathit{\tau }\right)-{\mathit{F}}_D,\hfill \\ \hfill & \mathit{\tau }={\mu }_{c}2\mathit{S},\hfill \\ \hfill & \mathit{S}=\frac{1}{2}\left({\displaystyle \frac{\partial V_i}{\partial x_j}}+{\displaystyle \frac{\partial V_j}{\partial x_i}}\right).\hfill \end{array}$$

(7)

In Equation (7), $\mathit{\tau }$$\left[Pa\right]$ is the viscous stress tensor, ${\mu }_c$$\left[\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}\right]$ is the dynamic coefficient of molecular viscosity of liquid phase, and $\mathit{S}$$\left[{\mathrm{s}}^{-1}\right]$ is the absolute value of the generalized velocity gradient.

2.3. Design and Procedure of Simulations

Figure 4 represents a sketch of the computational domain used in simulations.

In this liquid–particle model, we consider four types of boundary conditions, which are presented in

Table 1. Boundary conditions for numerical simulations.

Term	Method
Wall type boundary condition	No-slip condition
Symmetry type boundary condition	Impermeability condition
Inlet type boundary condition	Condition with a defined velocity field at the boundary
Free outlet type boundary condition	Condition with a zero static pressure gradient

.

The boundary condition on the left (see Figure 4) is normal for the boundary $V_{c,in}=V_{max}(1-{(y/R)}^2)$ for the fluid, where $V_{max}$ is the velocity at the center of the flow. The concentration of the dispersed phase, n, corresponds to a value of 0.074%. The boundary conditions on the right follows the zero gradient condition for static pressure, P, and dispersed phase concentration, n. We consider the upper boundary as a rigid wall with a no-slip velocity effect. The lower limit corresponds to a rigid wall without the no-slip effect. This approach is valid [15] and can be used for laminar flows in a straight axisymmetric channel. In this paper, the time step was calculated at each iteration based on the Courant–Friedrichs–Lewy condition (CFL) [52]. The shape of stenosis was determined by the proportion

$$y\left(x\right)=\left\{\begin{array}{cc}R\left[1-G\left\{L_{st}(x-L)-{(x-L)}^2\right\}\right],\hfill & L\le x<L+L_{st}\hfill \\ R,\hfill & x<L\mathrm{or}x>L+L_{st}\hfill \\ G=\frac{\left(R^2-y^2\left(x_{st}\right)\right)}{R^2},\hfill \end{array}\right.$$

(8)

and corresponds to the stenosis with overlap of $G=0.6$.

Parameters for the liquid–particle model are presented in

Table 2. Parameters for the liquid–particle model.

Parameter	Value	Unit
${\rho }_c$	1055	kg/m${}^3$
${\mu }_c$	0.004	Pa s
d (Aluminum particles)	20	µm
d (Calcite particles)	200	µm
$\rho$ (Aluminum particles)	2700	kg/m${}^3$
$\rho$ (Calcite particles)	2710	kg/m${}^3$
$m_d$ (Aluminum particles)	11.3 × 10${}^{-12}$	kg
$m_d$ (Calcite particles)	9.7 × 10${}^{-9}$	kg

.

To solve Equations (2)–(7), we use the finite-volume numerical scheme for a hexagonal mesh implemented in the FlowVision © CFD software (version 3.13.02) [53]. This software uses the original stable implicit velocity-pressure split algorithm to solve the momentum and pressure equations. In describing the particle dynamics, we used an Eulerian approach in which the dispersed and continuous phase interacted as interpenetrating continuous media. All computer simulations were executed on an HPC server with 2 × 64-core processors AMD EPYC 7763 (3.2 GHz).

Test calculations were performed to investigate mesh convergence and number of hexogonal cells. The initial grid was constructed to be non-uniform, with thickening in the vortex formation area (Figure 5), which allowed a clearer representation of the flow dynamics in the poststenotic region. The size of the minimum computational cell in the vortex area was 0.1 mm. The calculations were performed for a model with aluminum particles and the parameter of vortex area and mean velocity of particles was investigated; the results are presented in

Table 3. Results of the mesh independence study.

Mesh Size (k)	Mean Velocity Particles (m/s)	Vortex Area (m${}^2$)
117	0.00523	0.00016728
320	0.00542	0.0001616
414	0.00557	0.000162
850	0.00598	0.000162

and show that the difference between the results for the 414,000 and 850,000 mesh was less than 10%. This indicates that the chosen grid is sufficiently acceptable for this model.

3. Results and Discussion

3.1. Obtaining the Vortex Zone by Echo-PV Tracking of Calcite Particles

As was shown in Figure 2 (see Section 2.1), calcite particles are clearly visualized using medical ultrasound. The brightness of the particles by gray scale is 7–8 times greater than the brightness of the surrounding water (30–40 arbitrary units). In addition, the motion of particles in a fluid flow is also well determined. Figure 6 presents an example of the vortex zone visualization at the same experimental conditions. One can see the absence of particles in the upper part of the area behind the stenosis so that the shape and size of the zone are clearly visible.

In general, calcite particles are a well-suited substance for Echo-PV technology to evaluate the vortex zone in stenotic arteries. The high echogenicity of particles is ensured due to the optimal density and size of the material. Additionally, the particle size is at least an order greater than the dimensions of blood cells (∼10 µm). This implies no significant effects of particle–cell interaction on the trajectory of particle motion.

Recently, modern Ultrasound Localization Microscopy (ULM) technology with higher spatial–temporal resolution was introduced for Echo-PV. ULM operates with the scattered ultrasonic vibrations reflected from microbubbles used as an echo contrast agent [54]. The size of a single microbubble is in the range of 3–10 µm. ULM uses high-speed recording of reflected echo signals (up to 20,000 frames/s). Specially developed visualization algorithms are applied to obtain the location of each microbubble and their motion from frame to frame. As a result, ULM provides the spatial resolution of 5–20 µm.

The ULM method is a promising tool for in vivo measurements, in particular for microvascular mapping [55,56,57]. However, the motion of microbubbles depends on their mechanical interaction with blood cells. Entry of bubbles into a region with vortices and high shear stresses may result in the coalescence, fission, or dissolution of microbubbles, which potentially affects the accuracy of streamline detection in the vortex zones. Thus, the verification of ULM accuracy for the determination of local hemodynamic peculiarities is necessary. Perhaps the development of more complex models than those for solid particles will be needed for this purpose.

It is noteworthy that the limitations of the medical ultrasound used in this work in relation to the advantages of the ULM method for in vitro Echo-PV for vessels can be overcome by scaling up the vessel size of the phantom, maintaining the dimensions of the post-stenotic vortex zone. Further in vitro studies to confirm the feasibility of such a large-scale transition are planned.

3.2. Computer Simulations and Accuracy Investigation

The results of the comparison of simulation and experiment are summarized below.

Figure 7 shows the result of field simulations of 20-µm aluminum particles (as used in the experiment) and 200-µm calcite particles. The modeling results are superimposed on the photo of particle motion during the Azuma experiment [1]. We note a good qualitative correspondence of the distribution of the volume fraction of model particles with the density and trajectories of real particles.

However, the analysis of the velocities of model particles (Figure 8) shows that larger particles are not as well captured by the slow vortex behind the stenosis. The drag force grows slower than the inertia with increasing particle size. Therefore, the inertia of larger calcite particles begins to dominate and straightens the particle trajectory in the vortex zone. At the same time, the motion of small aluminum particles is to a greater extent determined by drag force, which entrains the particles following the streamlines inside the post-stenotic vortex zone. This leads to the circulation trajectories of the calcite particles in this region becoming indistinct. We can see this effect in Figure 6. At the same time, the small aluminum particles in the model give clear circulation streamlines, which agrees with the Azuma experiment.

A comparison of the particle velocity absolute value (Figure 9) shows that, regardless of the size, the particles are captured by the flow equally well (differences do not exceed 0.1% of the fluid velocity). However, in the poststenotic region (highlighted in the red box and magnified), the circulating vortex contours determined by the gradient method are significantly different.

To evaluate the quantitative difference between the methods using calcite and aluminum particles, we prepared a template of the measured parameters of the poststenotic zone. Figure 10 schematically shows the contour for the circulating vortices, which was determined by the gradient method (Figure 9).

Using this template, the results of the simulations were processed and the results were summarized.

Table 4. Comparison of the model with aluminum particles and the model with calcite particles. Parameters for comparison are presented in the scheme (see Figure 10).

Parameter (m)	Aluminum Particles	Calcite Particles	Difference (%)
$x_1$	0.0673	0.0679	0.88
$x_2$	0.094	0.096	2.08
Length	0.0267	0.0281	4.98
$y_1$	0.0038	0.00379	0.26
$y_2$	0.00395	0.00389	1.51
$y_3$	0.0020	0.0019	5

quantifies the poststenotic circulation region. The largest deviation is observed for the parameters “Length” and “$y_3$”. This is due to the fact that larger calcite particles have greater inertia and are not as well captured by the vortex. As can be seen, the use of large calcite particles for detection does not lead to a serious increase in the size of the vortex region (up to 5%), which is significant for the verification process. Note that the largest error is observed in determining the length of the vortex zone. This is explained by the effect of the greater inertia of large calcite particles, which, as was shown earlier, rarely get into the vortex zone and envelope it. Thus, this zone is closed later than when using small aluminum particles.

Experimentation and modelling allows us to formulate a number of limitations for the Echo-PV technique under consideration.

The method has significant dependence on the characteristics of ultrasonic equipment (low resolution and noise make it difficult to restore particle tracks and smear the boundary of the vortex zone), the liquid environment must be pre-cleaned and degassed to reduce noise, and the fluid pump must provide an even flow of fluid in the area of occlusion. For phantom vessels, this condition is not a big problem, but in the case of in vivo it is much more difficult to provide them.

Particle (marker) size should significantly exceed the size of blood cells and aggregates. If the size of the marker is comparable to the size of an erythrocyte, the track of the marker will be largely determined, not by the fluid flow, but by interaction with blood cells. For this reason, the PV technique with large (>100 µm) particles is more relevant for detecting flows in vessels. On the other hand, the use of large particles leads to a nonlinear dependence of the hydrodynamic and rheological forces on the velocity. Therefore, the accuracy of in vivo detection will strongly depend on the shape of the vessel, the properties of the liquid, and the properties of the ultrasound machine.

For asymmetric vessels, the choice of projection and depth of acoustic focusing will play a key role. It is likely that several measurements in different projections will be required. Finally, it should be taken into account that, especially in low-velocity fluid flows, the role of gravity will increase.

For in vitro experiments in phantom vessels, any available particle materials with sufficient echogenicity can be used. However, in vivo experiments require confirmation of the safety of materials and their concentration. It is very difficult to select large solid particles with such properties. In in vivo practice, gas bubbles are commonly used as tracers [58,59]. However, gas bubbles have several disadvantages. Solid particles have stronger acoustic backscattering and higher echo intensity compared to gas bubbles. This enhanced acoustic response allows for more reliable and accurate detection of solids, even under difficult flow conditions or in the presence of background noise. Gas bubbles, on the other hand, can have weaker acoustic signals, resulting in lower detection sensitivity and possible measurement errors. Particulate matter allows for better control of particle characteristics such as size, shape, and density that directly affect the accuracy of velocity measurements. Gas bubbles, due to their inherent variability in size and shape, can introduce uncertainty and errors in velocity measurements. Thus, additional theoretical and clinical studies are needed to select a particle material that has a sufficiently strong acoustic response, has stable and controllable characteristics, and can provide high accuracy in measuring localized vortex flows.

4. Conclusions

This paper presents a theoretical study of the Echo-PTV method for tracing streamlines and segmentation of the low-velocity vortex zone in vessels with stenosis. A mathematical model of the particle motion in a liquid flow was developed, and computer simulations were carried out for two experimental conditions. The first is the formation of a vortex zone by means of large-size calcite particles (100–200 µm in diameter) and the second—small-size microparticles of aluminum (20 µm in diameter) [1]. The possibility of calcite particles detection using the echolocation method was confirmed experimentally. Despite the fundamental difference between particle visualization methods (optical or Echo-PV), an error in determining the size of the vortex zone does not exceed 5%. Thus, calcite particles meet the requirements of Echo-PV technology for the in vitro verification of complex hemodynamic models.

Finally, this study is a first step towards applying the proposed methodology in clinical practice. Further in vitro studies are needed. We plan to compare the simulation results mentioned in this study with in vitro experimental findings that can be obtained using the Echo-PV methodology. In particular, the effects of various experimental variables (stenosis size and shape, flow rate, fluid viscosity, particle nature, etc.) on local flow features will be the focus of our next investigations.