Study on the radial sectional velocity distribution and wall shear stress associated with carotid artery stenosis

Atherosclerosis is an important cause of cardiovascular disease. The wall shear stress (WSS) is one of the key factors of plaque formation and dislodgement. Currently, WSS estimation is based on the measurement of the blood velocity gradient. However, due to the lack of flow field measurements in carotid stenosis vessels, the two distribution forms (parabolic and non-parabolic) commonly considered in numerical simulations could cause WSS estimates to differ by more than 40%, which could seriously affect the accuracy of mechanical analysis. This study applied three-dimensional (3D) printing technology to create an experimental model of real-structure carotid arteries. Microparticle image velocimetry was adopted to comprehensively measure blood velocity field data at the stenosis location, providing experimental validation of numerical simulation (Fluent; finite volume method) results. Then, the flow field was simulated at a normal human heart rate (45–120 beats per minute). The radial sectional velocity exhibited a plateau-like distribution with a similar velocity in the central region (more than 65% of the total channel width). This study provides an accurate understanding of the WSS at the carotid stenosis location and proposes a reliable method for the study of flow fields under various blood flow conditions.


Background
Atherosclerosis is an important pathogenic factor of cardiovascular disease. Formation of atherosclerotic plaques on the wall of blood vessels could lead to narrowing of blood ow pathways and even blood supply obstacles. Upon plaque dislodgement, serious consequences can occur, such as cerebral haemorrhage and stroke [1][2][3][4]. Animal experiments and clinical data studies have pointed out that haemodynamic characteristics, especially the blood ow wall shear stress (WSS), are key factors leading to arterial intimal thickening, plaque growth, and stroke [5].
At this stage, the WSS is indirectly calculated based on blood ow rate measurements. The peak blood ow velocity (V max ) in vivo has been measured mainly via magnetic resonance phase-contrast magnetic resonance imaging (PC-MRI) and ultrasound Doppler velocimetry (spatial resolution: 0.5-1 mm). Then, assuming that the velocity distribution in the blood vessel section is parabolic, the WSS has been indirectly calculated based on the maximum speed and vessel radius considering the law of Newton inner friction [6][7][8][9].
Therefore, whether the blood ow velocity conforms to a parabolic distribution is one of the key links in terms of the accuracy of WSS calculations. Due to the low spatial resolution (0.5-1 mm) of the PC-MRI and ultrasound Doppler velocimetry techniques, blood vessel-related ow eld information cannot be fully measured, so it remains di cult to reveal the characteristics of the ow velocity distribution [6, [9][10]. A higher-resolution velocimetry technique, namely, echo particle image velocimetry (echo-PIV, spatial resolution up to 0.5 mm), has been employed to study the ow eld distribution in a healthy carotid artery (diameter of 10 mm) in vitro, thus verifying parabolic distribution features [9]. In regard to carotid artery stenosis (only 1-3 mm in diameter) caused by plaques, this technology still cannot meet accuracy requirements. At present, there are no reports of direct measurements of the blood ow velocity eld at the location of carotid artery stenosis involving biological measurements or in vitro physical experiments.
Due to the lack of measured data, numerical simulation studies have highlighted two contradictory views, namely, in one view, the radial sectional velocity of blood owing across the stenosis location still exhibits a parabolic distribution [11], whereas the other view holds that a parabolic distribution form is not fully developed at the stenosis location [12]. WSS estimations based on these two distributions typically differ by more than 40%. Hence, it is necessary to carry out ow eld measurements at the location of vascular stenosis to accurately understand the characteristics of the ow eld.
To this end, in this study, a 3D printed model based on real con guration data was adopted to simulate the stenotic blood ow caused by carotid plaques, and a high-resolution ow velocity measurement technique (spatial resolution: up to 1 μm), i.e., microparticle image velocimetry (micro-PIV), was applied to conduct in vitro full velocity eld measurements to reveal the ow eld characteristics of carotid artery stenosis. On this basis, a numerical simulation study (Fluent; nite volume method) was carried out to reveal the sectional velocity distribution characteristics at the stenosis location. This study aimed (i) to provide support to accurately understand the WSS of carotid artery stenosis and (ii) to propose a reliable method for the study of the ow and shear elds under various blood ow conditions.

Materials And Methods
Three-dimensional reconstruction of carotid artery stenosis and measurement of the blood ow velocity and viscosity.
A computed tomography (CT) scanner (SOMATOM Spirit, Germany) was employed to obtain shape data pertaining to blood vessels near carotid plaques in clinical patients, and coronary and sagittal images of carotid arteries were generated in Mimics software (ver. 20.0) to construct a three-dimensional carotid artery vascular model. The resultant blood vessel model was smoothed (smoothing coe cient 0.85) in 3-Matic software (ver. 12.0). The average blood ow velocities at the entrance and two exits of the carotid artery model were measured with a colour Doppler ultrasound monitor (Figure 9 (a)). The patient blood viscosity measured with a blood viscometer reached 3.85 mPa·s.

Physical simulation experiment
In carotid artery stenosis blood vessel simulation model preparation, a 3D printer (Zhongrui SLA600, China) was employed to convert the determined three-dimensional geometric model of the blood vessel into a solid simulation model (a layer thickness of 0.1 mm, and the material was transparent photosensitive resin) (Figure 9 (b)).
The physical experiment method in this study relied on a peristaltic pump (YZ1515, China) (rotating speed: 100 r/min; ow rate: 276 mL/min) to provide the power required for uid ow. A latex tube (inner diameter: 5 mm) was adopted to connect the inlet of the experimental model to the peristaltic pump. A micro-PIV system (Lavision, Germany) was applied to realize velocity eld measurement. Fluoro-Max red uorescent dye (USA) with a peak particle size of 1 μm was employed to track uid ow. The experimental uid comprised a 65% dimethyl sulfoxide aqueous solution containing 0.3% uorescent particles. The viscosity was 3.72 mPa·s and the density reached 998.2 kg/m 3 . A device ow chart is shown in Figure 10. During the test, the experimental liquid was continuously injected into the model through the peristaltic pump. When bubbles were no longer observed in the experimental model, the micro-PIV system was activated to record velocity eld data, and Davis software (version 10.0.5) was used to process the measured data.

Numerical simulation calculation
In Fluent software (version 19.0), numerical calculations were performed to obtain a three-dimensional reconstruction of carotid artery stenosis, which was meshed as an unstructured tetrahedral mesh ( Figure  11 (a)), with a total of 250685 nodes and 1405028 meshes. The governing equation was the Navier-Stoke equation, and the ow mode was laminar ow. The inlet condition of the blood vessel was a function of the periodic velocity over time. The outlet was de ned as a free-ow outlet, and the relative pressure at each outlet was set to zero. Considering the physical experiment and the real heart rate cycle of the human body, the inlet cycle was set to range from 0.2-2.0 s. The aim was to analyse the ow pattern of the ow eld in the radial section (black line) at the location of carotid artery stenosis ( Figure 11 (b)). The ow channel width was normalized with a standardization-MinMax scaler.

Physical simulation results
In the physical simulation experiment, the ow eld velocity at the location of carotid artery stenosis was recorded through the micro-PIV system. When stable periodic ow was established, the maximum average velocity difference was smaller than 0.05 m/s. A velocity cloud chart and radial cross-sectional velocity distribution curve at the location of the vascular stenosis are shown in Figures 1 and 2, respectively.

Numerical simulation results
Veri cation of the numerical simulation accuracy In the numerical simulation, to verify the numerical calculation accuracy, the pulsation period was rst set to coincide with the peristaltic pump period (0.2 s). When stable periodic ow conditions were established, the maximum average velocity difference was smaller than 0.05 m/s. A velocity cloud chart and radial cross-sectional velocity distribution curve at the location of the vascular stenosis are shown in Figures 3 and 4, respectively.
The velocity distribution, maximum velocity, and velocity gradient near the vessel wall were compared between the numerical and physical experimental results. The velocity distributions determined with the above two methods all exhibited the form of a single-peak distribution. The ow velocities at the centre of the blood vessel were similar, and the velocity distribution indicated plateau-like features, with the width accounting for more than 65% of the total ow path. The deviation in the platform velocity measured with both methods was smaller than 20% at three different moments during the same cycle. The velocity gradients calculated based on the obtained data when the velocity signi cantly decreased were 1.5×10 3 and 1.75×10 3 s -1 , respectively. The numerical simulation accuracy was con rmed by the measured data.
Numerical simulation results under typical human heart rate conditions Combined with the real heart rate of the human body, the average ow velocity at the inlet was set to 0.6 m/s in the numerical simulations, and the period was 0.8 s. When stable periodic ow was established, the maximum average velocity difference was smaller than 0.05 m/s. The systolic velocity surface ( Figure 5), velocity cloud ( Figure 6) and ow velocity distribution curve (Figure 7) for the radial section at the stenosis location are shown below.

Discussion
In vitro physical simulation and measurement of the vascular ow eld Generally, there are two key aspects of in vitro experimental studies targeting the intravascular ow eld: production of the vessel structure model and measurement of the velocity eld within the vessel model.
First, the following two issues were included during modelling: the acquisition of vascular conformation data and the preparation of a solid blood vessel model. In terms of vessel con guration, early intravascular ow eld studies mainly employed simple ideal models. For example, simulation studies of the carotid artery relied on a simple Y-shaped structure, which was convenient but signi cantly differed from the actual structure and inevitably led to insu cient ow eld simulations [13][14]. For this reason, CT scanning techniques have been widely applied to obtain con guration information of real blood vessels and to accurately restore irregular vascular structures, such as intracranial aneurysms and carotid artery stenosis vessels [9,15]. In regard to intracranial aneurysms, relatively systematic numerical and physical simulation studies have been carried out based on the actual structure, revealing the effect of stent placement on the haemodynamics occurring within the aneurysm [16]. In terms of carotid stenotic vessels, haemodynamic analysis of the real vessel structure has only been performed via numerical simulation methods, but no physical simulation experiments have been conducted [17]. To transform 3D structural data into solid models suitable for physical experiments, currently, only 3D printing technology can be relied upon to achieve high-precision recovery (a printing accuracy of 0.05 mm) [18], which has been applied to the fabrication of ne structures such as heart and aortic valves [15], but the preparation of vascular models for carotid stenosis research has not been reported. Two types of materials can be chosen to prepare solid models via 3D printing: rigid materials (photosensitive resin, with an elastic modulus of 2000 MPa) and exible materials (hydrogel, with an elastic modulus of 80 kPa) [19][20][21].
Although the elasticity of the latter materials are close to that of carotid vessel walls (an elastic modulus of 47.7 kPa), the low transmittance (milky white colour) inhibits observation and measurement of the internal ow eld of the model [21][22]. Thus, photosensitive resin is currently the only viable model material to achieve optical measurements of the stenotic ow eld.
With regard to blood ow velocity eld measurements, the spatial resolution of the measurement technique should be compatible with the physical model scale. To perform ow eld analysis, at least 15 velocity data points are commonly measured in practice to yield a straight curve of the measured ow eld [23][24]. Therefore, techniques with a resolution higher than 40 μm are required at carotid stenosis sites. The commonly applied clinical ow velocimetry methods, PC-MRI and ultrasound Doppler velocimetry, cannot realize intravascular velocity eld measurements due to their low spatial resolution (0.5-1 mm) [6, 8-10, 25]. Echo-PIV with a higher spatial resolution (0.5 mm) relies on the microbubble contrast of tracer particles to measure the velocity eld. Measurement of the velocity distribution characteristics in the radial section of the common carotid artery (10 mm in diameter) has been achieved [8-10, 26], but the spatial resolution remains insu cient for the study of the velocity eld in stenosis vessels of the carotid artery (with an internal diameter of only approximately 1-3 mm). Currently, the micro-PIV technique with a maximum resolution of 1 μm can meet this requirement, which realizes the study of the velocity eld of blood cells in smaller arti cial microchannels (100 μm), yielding a parabolic distribution of the instantaneous velocity in the central plane [23,27]. This technique has not yet been applied in the study of carotid stenosis vessels.
Based on the above discussion, it can be deduced that the combination of 3D printing technology and the micro-PIV technique represents the only feasible technical route to realize ow eld measurement of carotid artery stenosis. In this study, this technical route was applied for the rst time to achieve ow eld measurement in stenotic vessels, and complete velocity eld data of blood owing across the stenosis location were measured within a ow eld of 1.5 mm 2 and 60 velocity points per unit length (1 mm) (in echo-PIV, only 2-3 data points are typically measured [8-9]), which notably improved the resolution and reliability of stenosis velocity measurement.
Radial cross-sectional velocity distribution characteristics of carotid artery stenosis.
The characteristics of the intravascular ow eld distribution are fundamental elements of WSS calculations [9,28]. Before this study, there were no reports on actual measurement of carotid artery stenosis or external ow eld measurement, which left the divergence (parabolic and non-parabolic) between the different velocity distributions in numerical simulations unresolved. In this study, physical experiments and numerical calculations were performed to clarify the characteristics of the radial crosssectional velocity distribution at the location of carotid stenosis: at normal human heart rates (heart rates ranging from 45-120 beats/min), the ow velocities in the central region of the stenosis location were similar and exhibited plateau-like distribution characteristics rather than an uncommon parabolic distribution.
From the available literature, as two common forms of the velocity distribution of blood ow, i.e., parabolic and plateau forms, the former form was mainly encountered in blood ow in healthy vessels and arti cial micro uidic channels [6, 8-9, 29], while the latter was mainly found in the blood ow eld of stenotic vessels with abrupt changes in vessel diameter [12]. Studies have analysed blood ow in stenotic vessels of carotid arteries via computer simulations, and the characteristics of the blood velocity distribution in the vessel radial cross-section were considered the result of incomplete blood ow development in the vessel [12]. Another numerical simulation study proposed that after blood ows across the location of blood vessel mutation, after a certain distance of a xed tube diameter (5 times the tube diameter length), the blood velocity distribution completely developed into a parabolic distribution [30]. Therefore, the sudden change in tube diameter caused by carotid plaques might be one of the important factors of the incomplete development of blood ow to form a plateau-like velocity distribution.

WSS estimation at the carotid artery stenosis location
Based on the characteristic plateau-like distribution of the velocity eld at the carotid stenosis location, the WSS was calculated as 20.35 Pa considering the maximum velocity gradient near the vessel wall. However, under the premise of a lack of clinical ow eld information, the WSS calculated based on the maximum velocity measured and the conventional parabolic distribution (the blue dotted line in Figure 8) [6-9] reached 7.25 Pa. Therefore, the current clinical estimation method of the WSS at the stenosis location could result in a difference up to 60%, which could notably affect the accuracy of mechanical analysis.
A full understanding of the distribution of the blood ow velocity eld is very important for a more accurate calculation of the WSS of carotid artery stenosis. However, carotid vessels of different shapes and varying degrees of stenosis might exhibit notably different ow eld characteristics [31-32], and clinical conditions do not facilitate comprehensive ow eld measurement. To solve this problem, it is necessary to carry out more basic research to clarify the relationship between the plaque shape, ow eld distribution and WSS, and amendments to the calculated WSS should be proposed for clinical stenosis research purposes to achieve accurate calculation.

Conclusions
In this study, based on carotid stenosis vessel data obtained from real patients, an experimental vessel model was fabricated with 3D printing technology for the rst time, and physical experiments were conducted by applying the micro-PIV technique to achieve full velocity eld measurement at the stenosis location. This study provides experimental validation of numerical simulation results and reveals the characteristics of the ow eld distribution at the stenosis location.
The results revealed that when blood owed across the carotid artery stenosis location, the velocity distribution was not parabolic but rather a plateau-shaped distribution, with a similar ow velocity in the central area (more than 65% of the total ow path). Therefore, the WSS values calculated based on a parabolic velocity distribution and the maximum velocity were nearly 60% lower.
This study provides a reliable method for WSS determination to better understand the vascular stenosis location and facilitate ow and shear force eld research. In the future, it is necessary to carry out indepth research on the relationship between the plaque shape, ow eld distribution and WSS, and amendments to the calculated WSS for clinical stenosis should be proposed.   Radial sectional velocity at the stenosis location under real human simulation conditions Figure 8 Distribution characteristics of the parabolic and platform velocity distributions based on the same maximum velocity Figure 9 (a) Three-dimensional reconstruction of a carotid artery stenosis vessel, and diameter and ow rate at the blood vessel inlet and outlet. (b) 3D printed simulation model.

Figure 10
Flow chart of the physical experimental device Figure 11 (a) Carotid artery model meshing, (b) carotid artery stenosis location radial section (black line)