Numerical investigation of arterial stenosis location affecting hemodynamics considering microcirculation function

Abstract

BACKGROUND:

In recent years, arterial stenosis has become one of the serious diseases threatening people’s life and health.

OBJECTIVE:

The main purpose of the present study is to examine the changes of hemodynamic parameters in different stenosis locations of arteries.

METHODS:

An arterial stenosis model with fluid-structure interaction and microcirculation as the outlet boundary of seepage is adopted in this paper. Considering the interaction between blood and arterial wall, a numerical simulation is carried out using the finite element method.

RESULTS:

The results show that hemodynamic parameters are sensitive to the change of stenosis location. The closer to the microcirculation zone the stenosis location, the lower the blood flow velocity, pressure and the wall shear stress. In addition, the velocity trend is transformed from the gradual increase to decrease with the increasing distance away from the inlet when the stenosis location moves to the microcirculation zone.

CONCLUSION:

This work proves that the stenosis location has a great influence on hemodynamics based on microcirculation function. Microcirculation is an important factor that cannot be ignored in the numerical simulation of arterial hemodynamics. The numerical results could provide the potential of clinical preconditions for disease diagnosis and treatment.

Keywords

Stenosis microcirculation hemodynamics fluid-structure interaction

1. Introduction

Arterial stenosis is a common atherosclerotic disease, which is caused by atherosclerotic plaque. With the improvement of people’s living standards, lifestyle and eating habits have gradually changed, thus, arteriosclerosis has gradually become the main cause of death [1]. Arterial stenosis has a great impact on the blood supply, and threatens people’s health. To date, the exact mechanism of the formation and development of arterial stenosis is not still clear. Many studies have reported a direct relationship between stenosis and hemodynamic parameters [2, 3, 4]. Studies have shown that microcirculation plays a key role in regulating blood flow. Therefore, it is necessary to incorporate microcirculation into the study of arterial hemodynamics [5]. In the process of research, many researchers can simulate the capillary structure by changing the porosity and permeability of porous media. Porous media can also well reflect the basic conditions of microcirculation seepage, and has been widely used [8]. To sum up the conclusions of previous studies, it is worth noting that most previous studies on arterial stenosis focused on the effect of stenosis degree, length or shape on blood flow [9, 10, 11]. However, there are relatively few studies on hemodynamics related to the arterial stenosis location. Studies on the effects of coronary artery stenosis on hemodynamics show that different stenosis locations have different effects on blood flow when microcirculation function is not considered [12]. Therefore, it is necessary to study the effect of stenosis location on hemodynamics considering microcirculation function.

Based on previous research, the microcirculation zone is designated as the outlet boundary condition according to the seepage theory. Combined with the fluid-structure interaction and microcirculation function, stenosis location is changed to investigate the changes of hemodynamic parameters. The purpose of this study is to compare the hemodynamic characteristics of different stenosis locations and improve our understanding of the pathogenesis of stenosis.

Figure 1.

The geometric model.

2. Materials and methods

2.1 The geometric model

Figure 1 shows the geometric model constructed in this paper, which is divided into artery zone and microcirculation zone, with the length of 10D and 2.5D respectively. The stenosis location is 6D away from the inlet. D is the diameter of the artery, D $=$ 20 mm and the thickness of the artery is 2 mm. The artery zone is set as a linear elastic wall. Due to the tiny deformation of the microvessels, it can be assumed that the microcirculation zone is a rigid wall. In this study, the definition formula of stenosis degree is as follows [13]:

$\displaystyle{S}_{0}=\frac{{R}_{0}-{R}_{\text{min}}}{{R}_{0}}\times 100{\%}$ (1)

where, ${R}_{0}$ is the radius of the uniform part of the pipe, ${R}_{\text{min}}$ is the smallest diameter in the stenosis zone. We study ${S}_{0}=50{\%}$ .

2.2 Numerical models and methods

2.2.1 The fluid model

We consider the unsteady flow in a compliant tube to simulate the pulsatile flow of arterial stenosis, and assume that the blood is an incompressible non-Newtonian viscous fluid. The artery zone is simulated by the interaction between blood flow and arterial wall. The governing equation of fluid can be written as Navier-Stokes equation of incompressible viscous fluid [14]:

$\displaystyle\frac{\partial{\rm{\bf u}}}{\partial{t}}+({({{\rm{\bf u}}-{\rm{% \bf u}}_{m}})\cdot\nabla}){\rm{\bf u}}=-\frac{1}{{\rho}_{f}}\nabla{p}_{d}+% \nabla\cdot({{\nu}({\nabla{\rm{\bf u}}+(\nabla{\rm{\bf u}})^{\rm{\bf T}}})})$ (2) $\displaystyle\nabla\cdot{\rm{\bf u}}=0$ (3)

Where, ${\rm{\bf u}}$ is the velocity of the fluid and ${\rm{\bf u}}_{{m}}$ is the grid velocity of the fluid, ${\rho}_{f}$ is the density of the fluid, ${p}_{d}$ is the flow pressure of the fluid, ${\nu}$ represents the coefficient of kinematic viscosity. This paper provides ${\rho}_{f}=$ 1050 kg/m ${}^{3}$ [15]. Carreau-Yasuda shear thinning model is used as the fluid model [16]:

$\displaystyle{\eta}={\eta}_{\infty}+({{\eta}_{0}-{\eta}_{\infty}})[{1+({{% \lambda\xi}})^{a}}]^{({{n}-1})/{a}}$ (4)

where, ${\eta}$ is the viscosity of the fluid. ${\eta}_{\infty}=2.2\times 10^{-3}\text{Pa}\cdot\text{s}$ is infinite shear viscosity, ${\eta}_{0}=22\times 10^{-3}\text{Pa}\cdot\text{s}$ is zero shear viscosity, ${\lambda}=$ 0.110 s is the time constant, ${n}=$ 0.392 is the power-law exponent, ${a}=$ 0.644, ${\xi}$ is the shear rate [17]. For a non-Newtonian fluid, ${\eta}$ is a function of ${\xi}$ . Porous media is composed of multiphase materials, which can simulate the microvascular tissue structure well. The controlling equation of blood flow in porous media is:

$\displaystyle\frac{\partial({{\varphi}{\rm{\bf u}}})}{\partial{t}}+{\rm{\bf u}% }\cdot\nabla({{\varphi}{\rm{\bf u}}})=-\frac{1}{{\rho}_{f}}\nabla({{\varphi p}% _{d}})+\nabla\cdot({{\varphi\nu}(\nabla{\rm{\bf u}}+({\nabla{\rm{\bf u}})^{\rm% {\bf T}}})})-\frac{{\eta\varphi}^{2}{\rm{\bf u}}}{{\rho}_{f}{K}}$ (5) $\displaystyle\nabla\cdot({{\varphi}{\rm{\bf u}}})=0$ (6)

${K}$ is permeability, and its expression is as follows:

$\displaystyle{K}=\frac{{d}^{2}{\varphi}^{3}}{180({1-{\varphi}})^{2}}$ (7)

where, ${d}$ is the average diameter of capillaries, we specify the porosity ${\varphi}=$ 0.5, ${d}=$ 8 $\mu$ m [18].

2.2.2 The wall model

In order to simplify the calculation, it is usually assumed that the arterial wall is isotropic, linear elastic and incompressible. We can calculate the total deformation and displacement of the arterial wall through the structural solver in the simulation software. The governing equation of the solid model is the constitutive equation of the linear elastic structure [19]:

$\displaystyle{\rho}_{S}\frac{\partial^{2}{d}_{i}}{\partial{t}^{2}}=\frac{% \partial{\sigma}_{{ij}}}{\partial{x}_{j}}+{F}_{i}$ (8) $\displaystyle{\rm{\bf\sigma}}={D}{\rm{\bf\varepsilon}}$ (9) $\displaystyle{\varepsilon}_{{ij}}=\frac{1+{\upsilon}}{{E}}{\sigma}_{{ij}}-% \frac{{\upsilon}}{{E}}{\sigma}_{{kk}}{\delta}_{{ij}}$ (10)

where, ${\rho}_{S}=$ 1120 kg/m ${}^{3}$ is the density of solid material, ${d}_{i}$ is the component of displacement, ${F}_{i}$ is the component of the force acting on the arterial wall, ${\sigma}_{{ij}}$ is the stress tensor, ${\varepsilon}_{{ij}}$ is the strain tensor, ${\delta}_{{ij}}$ is the Kronecker delta. In this paper, the elastic modulus ${E}=$ 5 MPa, Poisson’s ratio ${\upsilon}=$ 0.499 [21], indicating that the arterial wall is almost incompressible.

2.2.3 Boundary conditions and calculation methods

In order to solve the viscous flow field, the boundary conditions and initial conditions of the flow problem need to be given. In the transient structure solver, both ends of the artery zone are fixed, and the fluid-structure interaction interface is set to transmit displacement and pressure. The inlet velocity is set as shown in Fig. 2 [20], the flow cycle is 0.8 s, and the microcirculation seepage resistance is taken as the outlet boundary condition. The initial condition of the whole flow field is set to zero. Boundary conditions at the artery-microcirculation interface are given to guarantee the continuity of mass and pressure and the conservation of mass and momentum across the interface. At the same time, we set that there is no penetration and relative slip at the interface between the fluid and the solid. In other words, the velocity of the fluid and the velocity of the solid are equal, and the displacement of the fluid is equal to the displacement of the solid. The velocity of solid boundary is ${\rm{\bf U}}$ , which is expressed by the following formula:

$\displaystyle{\rm{\bf u}}={\rm{\bf U}}$ (11) $\displaystyle{\rm{\bf d}}_{\rm{\bf s}}={\rm{\bf d}}_{\rm{\bf f}}$ (12) $\displaystyle{\rm{\bf\sigma}}_{\rm{\bf s}}\cdot{\rm{\bf n}}_{\rm{\bf s}}={\rm{% \bf\sigma}}_{\rm{\bf f}}\cdot{\rm{\bf n}}_{\rm{\bf f}}$ (13)

where, ${\rm{\bf d}}_{\rm{\bf s}}$ and ${\rm{\bf d}}_{\rm{\bf f}}$ are the displacement vectors of solid and fluid respectively, ${\rm{\bf n}}_{s}$ and ${\rm{\bf n}}_{\rm{\bf f}}$ are the solid boundary normal vector and the fluid boundary normal vector respectively. The slip free condition can be expressed as:

$\displaystyle({{u},{v},{w}})|_{\Omega}=\frac{\partial{\chi}}{\partial{t}}$ (14)

where, $\Omega$ stands for the tube wall and ${\chi}=({{r},{\theta},{z}})$ is the location vector of the deformed tube wall.

Figure 2.

The inlet velocity.

We select a two-way fluid-structure interaction model. The finite element analysis of the arterial stenosis model is carried out. That is, based on the problem of each element, the above controlling equations are discretized by using the high-resolution advection scheme, and solved by the arbitrary Lagrangian-Eulerian algorithm [22, 23]. We use ANSYS to deal with the structural problem, while the transient behavior of the fluid domain is solved by CFX. The unstructured mesh consists of four node tetrahedral mesh in fluid domain and eight node hexahedral mesh in wall domain. During the calculation, the convergence criterion is root mean square (RMS) and it is set to $1\times 10^{-4}$ . The force generated by solid displacement and fluid flow is transmitted through the fluid solid coupling interface, and the calculation stops when it reaches the convergence standard. After the study of time step independence, the time step is set to 0.01 s, and three cycles are calculated to obtain the stable solution. The results of the third cycle are extracted for the analysis.

3. Results

The simulation results in this paper are obtained when the stenosis location is 6D away from the inlet. The results have been given in our previous work when the stenosis location is 5D away from the inlet [24]. In the following figures, 5D and 6D respectively denote that the stenosis location is 5D and 6D away from the inlet. Sections S1, S2, S3 and S4 are located at the same sites away from the inlet although the stenosis location is different. However, it should be noted that the distances from sections S1’ and S2’ to the stenosis location respectively are the same as those from sections S1 and S2 when the stenosis location is 5D away from the inlet.

3.1 Maximum velocity of different sections

When the stenosis location is 6D away from the inlet, Fig. 3 shows that the farther away from the inlet, the lower the velocity peak. Figure 3 also shows that the high flow velocity (6D) occurs at the upstream of the stenosis with a peak and it is approximately 0.65 m/s. The distribution and trend of maximum velocity are similar at sections S1 and S1’. At the downstream of the stenosis, the waveform of the velocity varies greatly at section S2’. Moreover, the velocity peaks of sections S2 (6D) and S2’ are below 0.6 m/s. The stenosis is prone to induce vortex flow. Thus, normal blood flow is disturbed. The vortex downstream of the stenosis significantly affects the velocity peak [25]. Therefore, the velocity peaks of S2 (6D) and S2’ have a large difference and a large phase difference. Figure 4 shows that the microcirculation zone has a very low flow rate and can be considered as steady flow.

Figure 3.

The maximum velocity distribution in the artery zone.

Figure 4.

The maximum velocity distribution in the microcirculation zone.

3.2 Maximum pressure of different sections

Figure 5 shows that the maximum pressure distribution and trend of sections S1 and S1’ as well as sections S2 and S2’ are basically the same. However, those of sections S2 and S2’ are 60 Pa or so, which indicates that there is a large pressure drop in the stenosis region (Fig. 6). Figure 5 also shows that the further away from the inlet, the lower the pressure peak and the higher the pressure bottom. In addition, the pressure values of sections S1 and S1’ are much larger than the pressure values of sections S2 and S2’. After 0.5 s, the pressure fluctuation in the artery zone tends to be stable. In the microcirculation zone, the distribution and trend of the maximum pressure on sections S3 and S4 in Fig. 7 are the same, and the corresponding pressure values are negative. The absolute value of pressure in section S4 is obviously larger than that in section S3. Similarly, there is also a large pressure drop in different locations of the microcirculation zone (Fig. 8).

Figure 5.

The maximum pressure distribution in the artery zone.

Figure 6.

The pressure drop distribution in the artery zone.

Figure 7.

The maximum pressure distribution in the microcirculation zone.

Figure 8.

The pressure drop distribution in the microcirculation zone.

3.3 Maximum wall shear stress of different sections

When the stenosis location is 6D away from the inlet, the wall shear stresses of sections S1 and S1’ are larger than those of sections S2 and S2’. Figure 9 shows that their distributions and trends are similar to those of sections S1 and S1’ in Fig. 3. Figure 10 shows that the change trend of the wall shear stress at sections S3 and S4 is consistent with that of the velocity shown in Fig. 4, and the values are very small. In addition, the wall shear stress at the downstream of the stenosis is relatively low.

Figure 9.

The maximum wall shear stress distribution in the artery zone.

Figure 10.

The maximum wall shear stress distribution in the microcirculation zone.

4. Discussion

By comparing these results with different stenosis locations, it is found that different stenosis locations have obvious effects on the distribution and trend of hemodynamic parameters in the artery zone. According to the velocity graphs in Fig. 3, it is found that when the stenosis location is 5D away from the inlet, the velocity gradually increases with the increasing distance away from the inlet, however when the stenosis location is 6D away from the inlet, it gradually decreases. The reason is that it is related to the microcirculation function. The velocity value of section S1 (6D) is obviously lower than that of section S1 (5D), which indicates that the blood flow velocity is reduced when the stenosis location is closer to the microcirculation zone. At the same time, because the stenosis location is very close to the microcirculation, the blood flow velocity cannot get rid of the influence of the microcirculation resistance reflection. As a result, when the stenosis location is 6D from the inlet, the velocity of section S2 in the downstream of the stenosis is lower than that of section S1. For different stenosis locations, the distribution and trend of blood flow velocity in the microcirculation zone are the same, and the velocity value is too small, which can be regarded as a steady state, as shown in Fig. 4.

The results when the stenosis location is 6D away from the inlet show that the loss of the role of microcirculation in regulating blood pressure in the stenotic artery leads to a gradual decrease in the range of pressure fluctuations. These results are similar to those when the stenosis location is 5D away from the inlet. They all violate physiological facts. The physiological facts of normal arterial circulation can be found in our previous work [26]. Combined with the analysis of the results of related blood pressure, it presents that no matter where the stenosis is in the artery, there is the unusual trend of the blood pressure. Due to the large pressure drop at the stenosis, low pressure occurs, which cannot provide sufficient blood flow power for the downstream and distal ends of the stenosis, that is, the microcirculation function is impaired. Compared with the case where the stenosis location is 5D away from the inlet, the amplitude of the maximum pressure of each section when the stenosis location is 6D away from the inlet is lower. The maximum pressure of section S1 (6D) in Fig. 5 is about 1200 Pa, and the peak pressure of section S1 (5D) is about 2000 Pa, which is much higher. Comparing the distribution trends of the maximum pressure in the microcirculation zone of the two stenosis locations, the pressures in Fig. 7 are all negative, and the absolute value of the maximum pressure (5D) is higher than the absolute value of the maximum pressure (6D).

Usually, high pressure easily induces high flow velocity and high wall shear stress. When the stenosis location is 5D away from the inlet, the results show that with the increase of blood flow velocity, high wall shear stress occurs at the downstream of the stenosis, as shown in Fig. 9. The so-called high shear stress is relative to the normal wall shear stress of human arteries. According to the literature, the average value of wall shear stress of different arteries is about 1.2 Pa [27]. However, when the stenosis location is 6D away from the inlet, due to the influence of the microcirculation resistance and wave reflection and the loss of microcirculation function, the blood flow velocity decreases and the wall shear stress is low at the downstream of the stenosis, as shown in Fig. 9. Under the action of high shear stress, the normal laminar flow of blood in the boundary layer near the blood vessel wall is destroyed, and complicated flow patterns may appear, which will lead to the hyperplasia of the vascular intima. Therefore, high shear stress contributes to the formation of arterial diseases. In general, the occurrence of arterial stenosis has a considerable relationship with microcirculation dysfunction. The wall shear stresses shown in Fig. 10 do not almost change due to the tiny values.

High pressure causes high blood flow velocity, high shear stress and low or negative pressure at the throat of the stenosis [28]. Stenosis may create significant flow resistance, large pressure drop [29]. Our results are found to be in excellent agreement with these facts. In the view of the degree, the far away from the microcirculation zone the stenosis location, the more disadvantageous it is.

These results are obtained through the establishment of an idealized model. Future work will be performed numerically to simulate real human arteries, so as to better develop diagnostic tools for vascular diseases and formulate appropriate treatment plans.

5. Conclusion

We numerically investigated the hemodynamic simulation of arterial stenosis based on the outlet boundary condition of the microcirculation, and discussed the influence of different stenosis locations on hemodynamic parameters in detail. Compared with previous studies, the results show that the location of the stenosis has a significant influence on the arterial blood flow velocity, pressure and wall shear stress. The completion of this work provides us with the favorable evidence to further understand the pathogenesis of stenotic arteries, and also provides certain guiding significance for early clinical diagnosis and treatment of arteries.

Footnotes

Conflict of interest

None to report.

Funding

This study is supported by the National Natural Science Foundation of China (81401492), the Science and Technology Project of Beijing Municipal Commission of Education (KM201510016012), the National Clinical Research Center for Cardiovascular Diseases, Fuwai Hospital, Chinese Academy of Medical Sciences (Grant No. NCRC2020007), and the CAMS Innovation Fund for Medical Sciences (2017-I2M-3-003).

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