Numerical analysis of hemodynamics in pulmonary artery stenosis

Abstract

BACKGROUND:

Pulmonary artery stenosis is a serious threat to people’s life and health.

OBJECTIVE:

The hydrodynamic mechanism of pulmonary artery stenosis is investigated.

METHODS:

Numerical analysis of hemodynamics in pulmonary artery stenosis using computational fluid dynamics techniques is performed. An idealized model of pulmonary artery stenosis is established, and the model is divided into main pulmonary artery, right and left pulmonary arteries, and their branches. The sections at different positions are intercepted to study the distribution trend of maximum velocity, pressure and wall shear stress.

RESULTS:

The numerical simulation results show that the pressure drop at both ends of the narrow area is large. High velocity and wall shear stress exist in the center of stenosis, and the wall shear stress at the distal end of stenosis gradually decreases, resulting in endothelial dysfunction.

CONCLUSIONS:

To some extent, this study helps clinicians make diagnosis and treatment plans in advance and improve prognosis. This method could be used in the numerical simulation of practical models.

Keywords

Pulmonary artery stenosis hemodynamics numerical simulation fluid-structure interaction

1. Introduction

Pulmonary artery stenosis is a common cardiovascular disease caused by congenital or atherosclerotic sclerosis and thrombolysis. Mild people will have symptoms such as palpitation and chest tightness after fatigue, and severe people will have complications such as hemoptysis and right heart failure [1]. Because both pulmonary stenosis and chronic thromboembolic disease cause the pulmonary vessels to be blocked or have poor blood flow, they can easily cause misdiagnosis in the clinic. Researchers and clinicians in various fields are committed to the combination of biomedicine and computer to improve the accuracy of arterial disease diagnosis. Studies have proved that the hemodynamics of main pulmonary artery and left and right branch arteries are conducive to the accuracy of pulmonary artery disease diagnosis [2,3]. In some cases, hemodynamic functional markers that cannot be detected clinically can be provided through modeling and simulation, such as wall shear stress, shear oscillation index and so on. These parameters can be calculated numerically using the patient’s specific arterial geometry and the mechanical properties between blood flow and arterial wall [4]. In recent years, there are more and more studies on computational fluid dynamics models of arterial stenosis, including the shape, length and degree of stenosis [5–7]. Zero/one-dimensional pulmonary artery model has been widely used in the clinic [8,9]. However, the zero/one-dimensional model is difficult to deal with the complex branches (bifurcation and bending parts) of pulmonary artery [10]. Bordones et al. [11] established a three-dimensional model and concluded that computational fluid dynamics can accurately simulate the hemodynamic characteristics of pulmonary circulation. Kong et al. [12] developed a highly parallel algorithm using a supercomputer with more than 10000 processors to get simulation of the unstable flow in the complete three-dimensional and patient specific pulmonary arteries, and the computational results were highly consistent with clinical markers of pulmonary vascular disease. Although the research has been recognized, it is difficult to realize this research in the clinic. It needs the blessing of supercomputer, which has great economic pressure and takes a long time. It is easy for patients to miss the best diagnosis and treatment time. In a study on pulmonary artery stenosis, Wan et al. [13] reconstructed the three-dimensional digital model of pulmonary artery according to CT data, and regarded the arterial wall as a rigid material. The simulation results obtained the distribution characteristics of blood flow velocity field, pressure and wall shear stress, which plays a reference role in evaluating the influence of different degrees of pulmonary artery stenosis and the selection of relevant surgical treatment schemes. Karayannacos et al. [14] and Andayesh et al. [15] studied the effect of inter stenosis distance and shape size on arterial hemodynamics, respectively, and both concluded that there was a large pressure drop downstream of the stenosis. Some research also considered the interaction of blood flow with the arterial wall, and a two-dimensional problem of non-steady blood flow was solved using CFD-ACE in the study by Spilker et al. [16]. Nowadays, researchers and clinicians are committed to analyzing the relationship between pressure distribution, velocity distribution and wall shear stress in patients through numerical simulation, so as to optimize the diagnosis and treatment scheme. In the real clinical diagnosis and treatment, most clinicians invasively measure the pressure at both ends of the stenosis by inserting a pressure guide wire. For serious patients or patients with very poor physical quality, invasive methods will seriously threaten the life and health of patients. Numerical simulation technology is non-invasive, intuitive and effective, so it is considered to be an important tool to study cardiovascular diseases.

At present, the numerical simulation analysis of pulmonary artery hemodynamics of two and three branches has been mostly investigated, and there is little research on four pulmonary artery branch models [17,18]. In this paper, an idealized model of four branches of pulmonary artery with stenosis is established. In patients with pulmonary artery stenosis, the blocking site often appears on the left and right pulmonary artery branches. Therefore, we make a stenosis structure on the left pulmonary artery branch. The interaction between blood flow and vascular wall deformation is studied by constructing an idealized model, so as to study the hemodynamic characteristics of human pulmonary artery. This work mainly includes the changes of blood flow maximum velocity, pressure and wall shear stress, and then we analyze the effects of these changes on the physiological structure or function of human pulmonary artery. With the continuous development of computer and medical imaging technology, the numerical simulation results of pulmonary artery stenosis provide new means and ideas for clinical diagnosis and treatment. It is possible to make an accurate diagnosis of patients quickly under noninvasive conditions.

2. Materials and methods

2.1. The geometric model

In this paper, we simplify the pulmonary vasculature into a combination of smooth curved arterial geometries based on the physiological structure of the human pulmonary artery and the structure of each branch, and the diameter as well as the length of each branch are within the human true pulmonary artery size range [19,20]. As shown in Fig. 1, the length of the main pulmonary artery (MPA) is 50 mm and the diameter is 24 mm, the horizontal length of the right pulmonary artery (RPA) is 60 mm and the diameter is 20 mm, the horizontal length of the left pulmonary artery (LPA) is about 75 mm and the diameter is 18 mm, the horizontal length of the right front end artery (RTA) is about 18 mm and the diameter is 10 mm, and the length of the left front end artery (LTA) is about 70 mm and the diameter is 8 mm. The wall thickness is 1 mm. In order to simplify the calculation, a linear elastic model is adopted for the wall. The stenosis degree of 50% is set in the left anterior artery branch, and the calculation formula of stenosis degree is given [21]: $\begin{eqnarray}\displaystyle \text{S}_{0}=\frac{\text{R}_{0}-\text{R}_{\min }}{\text{R}_{0}}\times 100\text{% }. & & \displaystyle\end{eqnarray}$ (1) Where, R ₀ represents the radius of the uniform part of the pipe, and R _min represents the smallest diameter within the narrow area.

Fig. 1.

The idealized model of the pulmonary artery.

In this model, the cylindrical coordinate system is adopted, and the entrance center is taken as the origin. In order to fully observe the hemodynamic changes of pulmonary artery and its branches, sections are intercepted on each branch. S1 is 25 mm away from the inlet. S2 is 60 mm away from the outlet 3. S3 is 40 mm away from the outlet1. S4 is 20 mm away from the outlet 2. S5 and S6 are respectively located at 30 mm upstream and downstream of the stenosis. In this work, we focus on the trends of pressure, wall shear stress and velocity in different sections.

2.2. The numerical model

Blood produces force interaction in the process of vascular flow, which follows the hemodynamic equation, vascular wall dynamics equation and fluid-structure interaction condition. For solids, the incompressible, isotropic and linear elastic model is adopted for the wall. Solid density 𝜌_S = 1120 kg/m³ [22], elastic modulus E = 2 MPa [23], Poisson’s ratio 𝜈 = 0.49 [12]. By using this simple model, we just want to observe and analyze the changes of pressure, velocity and wall shear stress during the interaction between pulmonary artery wall and blood flow in a short time. The constitutive equation, strain displacement equation and the relationship between stress and strain of isotropic linear elastomer are as follows [24]: $\begin{eqnarray}\displaystyle {\sigma}_{ij}\text{} & =\text{} & \displaystyle {\lambda}{\varepsilon}_{ij}{\delta}_{ij}+2\text{G}{\varepsilon}_{ij}\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle {\varepsilon}_{ij}\text{} & =\text{} & \displaystyle \frac{1}{2}(d_{i,j}+d_{j,i})\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle {\varepsilon}_{ij}\text{} & =\text{} & \displaystyle \left(\frac{1+{\nu}}{\text{E}}\right){\sigma}_{ij}-\frac{{\nu}}{\text{E}}{\sigma}_{kk}{\delta}_{ij}.\end{eqnarray}$ (4) Where 𝜎_ij is the stress component, 𝜆, G are called lame elastic constants, 𝜀_ij is the strain component, 𝛿_ij is Kronecker Delta, d _i and d _j represents the displacement in the i and j directions respectively, ⋅, i and ⋅, j represent the derivatives of the i and j variables respectively.

For fluids, fluid density 𝜌_f = 1060 kg/m³ [17], dynamic viscosity 𝜇 = 0.004 Pa.s [17], using incompressible, unsteady Newtonian fluid. According to the analysis of the maximum velocity of systolic inlet blood flow and the maximum diameter of pulmonary artery, the Reynolds number is about 4000, therefore we set the blood flow as turbulent. The hemodynamic equations are as follows [25]: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\rho}_{\text{f}}\left[\frac{\partial \mathbf{u}}{\partial t}+((\mathbf{u}-\mathbf{u}_{\mathbf{m}})\cdot {\nabla})\mathbf{u}\right]=-{\nabla}p_{d}+{\mu}{\nabla}^{2}\mathbf{u}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\cdot \mathbf{u}=0.\end{eqnarray}$ (6) Where u is the fluid velocity, u _m is the mesh velocity of the fluid, p _d is the fluid flow pressure.

2.3. Boundary conditions

In pretreatment, we need to set inlet, outlet and wall boundary conditions. The inlet velocity is shown in Fig. 2 and the data points are fitted with Fourier function by MATLAB. The inlet fitting formula is: $\begin{eqnarray}\displaystyle f(x)=a_{0}+a_{1}\ast \cos ({\omega}\ast x)+b_{1}\sin ({\omega}\ast x)+a_{2}\ast \cos (2{\omega}\ast x)+b_{2}\ast \sin (2{\omega}\ast x). & & \displaystyle \nonumber\end{eqnarray}$ It is obtained by Fourier function fitting that a ₀ = 0.1873, a ₁ = −0.1411, a ₂ = −0.05947, b ₁ = 0.2342, b ₂ = −0.09533, 𝜔 = 9.277. Therefore, the period of inlet velocity $T=∼\frac{2{\pi}}{{\omega}}\approx 0.7$ s.

Fig. 2.

The inlet velocity.

The outlets are set to zero static pressure and the initial conditions are all set to 0. During the calculation, all ports of the vessel are constrained. In this study, two-way fluid solid coupling is adopted, and the interface between fluid and solid is set to be no slip condition. The slip free condition can be expressed as: $\begin{eqnarray}\displaystyle (u,v,w)|_{{\Omega}}=\frac{\partial {\chi}}{\partial t}. & & \displaystyle\end{eqnarray}$ (7) Where, Ω stands for the tube wall and 𝜒 = (r, 𝜃, z) is the location vector of the deformed tube wall.

2.4. The computational methods

In this study, different commercial solvers are used to calculate the structure and fluid. ANSYS transient structure analysis is used for solid analysis, and CFX solver is used for fluid calculation. The two solvers are connected through modules for data transmission. All equations are coupled into a system and discretized in space in turn, which is solved by arbitrary Lagrange Euler algorithm. The data generated by the interaction between fluid and solid are transmitted through the fluid solid coupling interface. In order to obtain the periodic steady-state solution, three cycles are calculated, and the results of the third cycle are extracted and used [26]. The total calculation time is 2.1 s. The sensitivity analysis of the selection of time step is carried out, and the appropriate time step is 0.01s. We set the convergence criterion 1 × 10⁻⁴ (RMS) to control the number of iteration steps. In the process of numerical simulation, mesh generation is very important. In order to ensure the mesh quality and simulation accuracy, the mesh correlation test is carried out. Finally, tetrahedral grids are used for both solid and fluid grids, of which the total number of solid grids is 64360 and the total number of fluid grids is 91731.

3. Results

3.1. Maximum pressures of different sections

Figure 3 depicts the maximum pressure variation curves for different sections. The pressure fluctuation range of section S1 on the main pulmonary artery is from −700 to 800 Pa, and the absolute value of pressure is greater than that of other sections at each time. The pressures of sections S2, S3, S4 and S5 are between −300 and 400 Pa, and the differences of pressure values are between 0 and 300 Pa. Section S6 is close to the outlet 4, therefore the pressure value is about 0. The pressure greatly fluctuates during systole, and the pressure waveform changes little during diastole, and gradually tends to be stable. It can be seen from the pressure trends of sections S5 and S6 that the pressure at the distal end of the stenosis is much lower than that at the proximal end of the stenosis, indicating that there is a large pressure drop at the stenosis. The pressure drop diagram is plotted in Fig. 4, and the pressure difference is between −300 and 350 Pa.

Fig. 3.

Maximum pressures of different sections.

Fig. 4.

Pressure drop at both ends of stenosis.

3.2. Maximum velocities of different sections

Figure 5 shows the maximum velocity distributions and trends of the six sections. The velocity distributions at sections S1, S2 and S3 are consistent with the inlet velocity distribution. The peak values at sections S1, S2, S3 and S4 occur at 0.23 s during ventricular contraction. At the end of systole, the velocity gradually decreases and tends to be stable and ready for the beginning of the next cycle. Due to the large difference in arterial diameter, there is less shunting of blood into the small branches of left and right pulmonary arteries, and the velocities at sections S4, S5 and S6 decrease temporarily. The velocities at sections S5 and S6 have two peaks, which reach at 0.23 and 0.45 s respectively, and then gradually tend to a stable state. The peak velocity of the main pulmonary artery is about 0.7 m/s and it is higher than that of the left and right branch arteries. There is little difference between sections S5 and S6. In order to more intuitively observe the blood flow, the velocity streamlines are derived in Fig. 6. Four typical moments, namely 0.10, 0.23, 0.40, 0.60 s are selected, representing the pre-systole, mid-systole, pre-diastole, post-diastole. At any time, the stenosis shows high flow velocity, and the blood flow in the main pulmonary artery is more smooth and rapid. In the late diastolic stage, the blood flows into the left and right branches and bifurcation of pulmonary artery in a vortex state, and there is a recirculation fluid area.

Fig. 5.

Maximum velocities of different sections.

Fig. 6.

Velocity streamlines.

3.3. Maximum wall shear stresses of different sections

It can be seen from Fig. 7 that the wall shear stress distributions and trends of sections S1, S2, S3 and S4 are similar to the velocity, and all reach the peak at 0.23 s. Sections S5 and S6 have peaks at 0.23 and 0.45 s. The peak value of wall shear stress of section S1 reaches 1.5 Pa, and it is higher than that of other sections, and the peak values of other sections are between 0.45 and 1 Pa. Similarly, the contours of wall shear stress during different periods are derived. Figure 8 depicts that there is a high wall shear stress in the narrow area and a low wall shear stress at both ends of the narrow area.

Fig. 7.

Maximum wall shear stresses of different sections.

Fig. 8.

Contours of wall shear stress.

4. Discussion

The blood pressure fluctuation range is the largest on the normal main pulmonary artery branch, which is in line with the physiological fact [27]. However, due to stenosis, the pressure fluctuation on the left and right pulmonary branches gradually decreases, which is contrary to the physiological facts. Stenosis will produce flow obstruction and thus there is a large pressure drop at both ends of the stenosis, which indicates the occurrence of disease [28]. It is observed from Fig. 5 that the peak pressure of section S1 is greater than the peak inlet velocity, the blood flows into other branches after entering the main pulmonary artery, and the diameter of the branch artery is smaller than that of the main pulmonary artery. Therefore, the blood flow velocity increases without reducing the blood flow. It is well known that stenosis hinders blood flow and leads to reflection, resulting in peak superposition. Therefore, the velocity curve of the section at both ends of the stenosis shows a transient increase in blood flow at 0.45 s. The accuracy of the numerical simulation in this paper is verified by comparing with the results of Plourde et al. [28]. It can be seen from Fig. 6 that the blood flow is disturbed. There is vortex at the intersection of curved arteries, resulting in the damage to endothelial cells. In addition, the vortex state of low flow rate will lead to the decrease of blood vessel elasticity and the increase of blood viscosity, and the insufficient blood supply is prone to appear in human body. The blood flow velocity in the stenosis center is higher, the blood flow velocity in the distal stenosis decreases gradually, and the wall shear stress also changes. Wall shear stress plays an important role in the pathophysiology of vascular remodeling by regulating the growth and proliferation of endothelial cells [29]. During systole, the blood flow velocity is high, and there is a high wall shear stress in the bifurcation and stenosis center, which is related to the rapid impact of blood flow in the stenosis center. Because the distal end is affected by stenosis, the uneven blood flow velocity and turbulence gradually reduce the wall shear stress at the distal end of stenosis, which can lead to the increase of pulmonary artery pressure [30]. In addition, the blood flow at low wall shear stress is often stagnated and the residence time of blood flow is prolonged. The endothelial cells of the vascular wall absorb more side flow ions in the blood and form a blockage [31]. Many studies have shown that low wall shear stress is conducive to the development of atherosclerosis, and high wall shear stress will inhibit the degradation of endothelial cells and effectively avoid atherosclerosis [32,33]. However, when the wall shear stress is higher than a certain range, the plaque will rupture. Through this study, it is found that atherosclerotic plaque may rupture or develop in a bad direction in the central area of stenosis. The low wall shear stress between the proximal and distal end of stenosis may contribute to the severity of atherosclerotic sclerosis in the area around stenosis. Therefore, high wall shear stress is helpful to the development of pulmonary artery disease. In this study, the low wall shear stress at the distal end of the stenosis may promote the formation and development of thrombosis with the increase of the degree of stenosis [34]. The contribution of this study shows that abnormal pulmonary hemodynamics can be used as an early warning of disease and it provides the guided significance for clinical noninvasive diagnosis of pulmonary artery stenosis.

With the continuous development of computer technology and medical biomechanics, numerical simulation technology has been widely used in cardiovascular diseases. However, there are inevitable limitations in this paper. In the condition setting of hemodynamic simulation, Newtonian fluid is used for blood, isotropic linear elastic model is used for arterial wall, and the taper of arterial wall is ignored. The setting of the inlet and outlet of blood vessels and the permeability of blood vessels are different from the real physiological environment. In future research, the pulmonary artery model of real patients can be considered to improve the accuracy of the model and the authenticity of numerical simulation results. Therefore, it should be noted that this study is a methodology paper, which lays a foundation for the later numerical simulation of pulmonary artery in specific patients.

5. Conclusion

Based on the establishment of an idealized model of pulmonary artery stenosis, the changes of maximum velocity, pressure and wall shear stress on different sections of four branches of pulmonary artery are numerically simulated. Through the curve distributions of hemodynamic parameters, it can be concluded that the abnormality of hemodynamic parameters can be used as a new basis for the diagnosis and treatment of pulmonary vascular diseases, which can help us better study the formation and development mechanism of diseases such as pulmonary artery atherosclerosis, and serve the prevention, clinical diagnosis and treatment of this disease. The research method in this paper can be extended to more branch artery models or real CT data modeling. Thus, we could have a deeper understanding of the internal physiological structure and mechanical properties of arteries. It is a favorable tool for making treatment plans. This study helps clinicians to better understand the impact of pulmonary artery stenosis on hemodynamics. Thus, adequate attention and timely intervention may be able to prevent further deterioration of arterial stenosis and improve prognosis.

Footnotes

Acknowledgements

The authors thank the National Natural Science Foundation of China (Grant no. 81401492) for financially supporting this research. The work is also supported by the National Clinical Research Center for Cardiovascular Diseases, Fuwai Hospital, Chinese Academy of Medical Sciences (Grant no. NCRC2020007) and CAMS Innovation Fund for Medical Sciences (Grant no. 2017-I2M-3-003).

Conflict of interest

None to report.

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