A hemodynamic model of artery bypass graft considering microcirculation function

Abstract

BACKGROUND:

The incidence of arterial stenosis is increasing year by year. In order to better diagnose and treat arterial stenosis, numerical simulation technology has become a popular method.

OBJECTIVE:

A novel model is constructed to investigate the influence of microcirculation on the hemodynamics of artery bypass graft.

METHODS:

In this paper, a severely narrow artery bypass graft model is considered. The geometric shape includes a narrow artery tube and a bypass graft of the same diameter with a 45° suture angle. The fluid-structure interaction model is considered by finite element numerical calculation, and the flow is simulated with microcirculation as the outlet boundary condition. The changes of blood flow velocity, pressure and wall shear stress are analyzed.

RESULTS:

The results show that blood almost entirely flows into the graft tube and there is no recirculation area at the anastomosis.

CONCLUSION:

The artery bypass graft model considering microcirculation function could simulate the physiological characteristics of blood flow more reasonably, and it provide helps for clinicians to diagnose and treat arterial stenosis.

Keywords

Stenosis microcirculation artery bypass graft hemodynamics fluid-structure interaction

1. Introduction

Atherosclerosis is a common chronic vascular disease. Cholesterol and some low-density lipoproteins accumulate in the lining of blood vessels that thicken the walls and make their elastic properties lose. When the substances continue to accumulate in the inner wall of blood vessel, it may partially or completely block the blood flow in the tube [1]. This abnormal accumulation of lipids is commonly referred as stenosis, which could often be associated with the formation of blood clots, atherosclerosis and rupture of plaque caps. Atherosclerosis is a persistent narrowing during the heart cycle that has a certain impact on the hemodynamics [2,3].

Atherosclerotic plaque in the coronary arteries may cause a blockage of blood flow to the heart, which could lead to myocardial ischemia. Arterial stenosis generally occurs in important arteries of the human body, such as coronary arteries, carotid arteries, vertebral arteries, and hepatic arteries, which will lead to blood disorders in the downstream artery region [4,5]. For the treatment of arterial stenosis, according to different severities of stenosis, it could be divided into drug therapy, interventional therapy and artery bypass graft surgery. Artery bypass graft is a form of treatment in which bypass grafts are implanted into a narrow host artery, allowing blood flow to bypass the blocked area and be supplied directly to the distal end of the artery to improve blood flow in the body [6,7].

In the past, in order to study the hemodynamic changes in bypass vessels, many researchers used the method of three-dimensional modeling to establish an idealized artery model and conduct numerical simulation of artery bypass graft. In addition, most previous studies on artery bypass graft focused on the suture angle, geometric shape and graft characteristics, including numerical studies on single, double and triple artery bypass graft [8–10]. Some scholars [11] studied the influence of the characteristics of graft tubes on the hemodynamics, conducted fluid-structure interaction analysis of arteries, and studied the bypass performance of the thoracic artery and saphenous vein grafts. It is concluded that wall shear stress of the saphenous vein is higher than that of the thoracic artery, and the total deformation of the thoracic artery is greater than that of the saphenous vein, suggesting that venous graft and lower degree of stenosis are more critical in restenosis.

In current research, some idealized models of artery bypass graft ignore the influence of capillaries and other arterial tissues, which may lead to inaccurate numerical results. Microcirculation is an important part of blood circulation system, which could regulate blood flow and tissue perfusion and affect blood pressure [12]. The flow in microvascular tissue could be considered as the seepage in porous media. In previous studies, the hemodynamic model considering microcirculation has been numerically simulated, and the microcirculation zone is set as the outlet boundary condition, which provides the valuable study of hemodynamics under physiological and realistic boundary conditions [13]. Xu [14] established a symmetric coronary artery stenosis model, conducted three-dimensional simulation of blood flow, applied a seepage outlet boundary condition, and discussed the influence of microcirculation on the fractional flow reserve (FFR). It is found that FFR increases under the influence of microcirculation resistance. That is, microcirculation has a strong hemodynamic effect on coronary arterial stenosis.

However, there are few numerical simulations of artery bypass graft with microcirculation. It is necessary to include microcirculation in the study of hemodynamic simulation of artery bypass graft. This purpose of this study is to explore the effects of microcirculation on hemodynamics of artery bypass graft by constructing a three-dimensional idealized model. The focus is to simulate the microcirculation system using porous media, which could reflect the percolation characteristics of microcirculation. It could improve the accuracy and effectiveness of hemodynamic modeling.

2. Materials and models

2.1. Geometric model

Figure 1 shows the geometric model constructed in this study. It is an idealized model based on a range of mean physiological values for the size of normal adult coronary arteries. The model simplifies the geometry of real blood vessels and removes some local features of blood vessels. It is divided into two parts, including the artery and microcirculation zones, with the lengths of 137 and 25 mm respectively. The stenosis exists in the 60 mm position away from the entrance. The geometric shape includes a host artery tube and a bypass graft tube of the same diameter with a 45° suture angle. The arterial diameter is 4 mm and the wall thickness is 0.5 mm. Due to the tiny deformation of microvessels, microcirculation is set to be rigid. In this study, the degree of stenosis is defined as $\begin{eqnarray}S_{0}={\displaystyle \frac{R_{0}-R_{\min }}{R_{0}}}\times 100\%.\end{eqnarray}$ (1) Where, R ₀ is the radius of the uniform part of the blood vessel, and R _min is the minimum radius of the narrow area. In this study, we set S ₀ = 90%.

Fig. 1.

The geometric model.

2.2. Numerical model and method

2.2.1. The fluid model

We set blood as an incompressible non-Newtonian viscous fluid. The interaction between blood flow and the deformation of the arterial wall in the artery zone is adopted. The governing equations of the fluid are as follows. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\rho}_{f}\left(\frac{\partial \mathbf{u}}{\partial t}+((\mathbf{u}-\mathbf{u}_{m})\cdot {\nabla})\mathbf{u}\right)=-{\nabla}p_{d}+{\nabla}\cdot ({\eta}({\nabla}\mathbf{u}+({\nabla}\mathbf{u})^{T}))\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\cdot \mathbf{u}=0.\end{eqnarray}$ (3) Where, u represents the fluid velocity, u _m represents the mesh velocity, 𝜌_f refers to the fluid density, and P _d is the fluid pressure. In this study, we set 𝜌_f = 1060 kg∕m³ [15 ]. Blood behaves non-Newtonian and we adopt the Carreau-Yasuda shear thinning model. $\begin{eqnarray}{\eta}={\eta}_{\infty }+({\eta}_{0}-{\eta}_{\infty })[1+({\lambda}{\xi})^{{\alpha}}]^{(n-1)/{\alpha}}.\end{eqnarray}$ (4) Where, 𝜂 represents the blood viscosity, 𝜂_∞ is infinite shear viscosity, 𝜂₀ is zero shear viscosity, 𝜆 is the time constant, 𝜉 is the shear rate, 𝛼 is the power law index, and the value of n determines the properties of the fluid. In this paper, we set the above parameters to: 𝜂_∞ = 2.2 ×10⁻³ Pa ⋅ s, 𝜂₀ = 22 ×10⁻³ Pa ⋅ s, 𝜆 = 0.110 s, n = 0. 392, 𝛼 = 0.644 [16].

Porous media are composed of a large number of dense groups of small voids, and play an important role in simulating the microstructure of microvessels. We treat the microcirculation zone as porous media. The governing equations of blood flow in the zone are as follows. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\rho}_{f}\left({\displaystyle \frac{\partial ({\varphi}\mathbf{u})}{\partial t}}+\mathbf{u}\cdot {\nabla}({\varphi}\mathbf{u})\right)=-{\nabla}({\varphi}p_{d})+{\nabla}\cdot ({\varphi}{\eta}({\nabla}\mathbf{u}+({\nabla}\mathbf{u})^{T}))-{\displaystyle \frac{{\eta}{\varphi}^{2}\mathbf{u}}{K}}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\cdot ({\varphi}\mathbf{u})=0.\end{eqnarray}$ (6) Where, K represents the permeability of porous media, it is calculated as follows: $\begin{eqnarray}K={\displaystyle \frac{d^{2}{\varphi}^{3}}{180(1-{\varphi})^{2}}}.\end{eqnarray}$ (7) Where, d represents the diameter of the capillary, 𝜑 is the porosity. In this paper, we set d = 8 μm [17] and 𝜑 = 0.5 [18].

2.2.2. The arterial wall model

For the arterial wall, we assume an isotropic, incompressible linear elastic model for simple calculation. We set solid density 𝜌_s = 1120 kg/m³, elastic modulus E = 5 MPa, Poisson’s ratio 𝜇 = 0.499 [19]. The governing equations of the wall are as follows. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\rho}_{s}{\displaystyle \frac{\partial ^{2}d_{i}}{\partial t^{2}}}={\displaystyle \frac{\partial {\sigma}_{ij}}{\partial x_{j}}}+F_{i}\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\varepsilon}_{ij}=\left({\displaystyle \frac{1+{\mu}}{E}}\right){\sigma}_{ij}-{\displaystyle \frac{{\mu}}{E}}{\sigma}_{kk}{\delta}_{ij}.\end{eqnarray}$ (9) Where 𝜎_ij represents the stress component, 𝜀_ij represents the strain component, F _i is the component of the body force acting on the wall, d _i is the displacement in the ith direction, 𝛿_ij is the Kronecker delta.

2.2.3. Boundary conditions and computational method

The tetrahedral mesh is selected for the fluid, and the hexahedral mesh is used for the wall. The mesh independence analysis is carried out. After repeated experiments, the appropriate mesh size is selected. The solid and fluid meshes are divided into 26619 and 48908 cells. ANSYS transient structure and CFX modules are used to deal with the solid and fluid domains. In the process of numerical simulation, the two-way fluid-structure interaction model is adopted. In the transient structure solution, the two ends of the artery zone are fixed, and the fluid-structure interface is set to transmit pressure and displacement. We set the inlet velocity, as shown in Fig. 2 [20]. The microcirculation zone is set as the seepage outlet boundary condition, and the initial condition of the whole flow field is set to be zero. At the interface between the artery and microcirculation zones, the continuity of mass and pressure and the conservation of mass and momentum are guaranteed. At the same time, it is set to be no-permeable and no-slip, and the equations at the fluid-structure interface are as follows. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle u=U\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle d_{s}=d_{f}\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\sigma}_{s}\cdot n_{s}={\sigma}_{f}\cdot n_{f}.\end{eqnarray}$ (12) Where, u and U represent the fluid velocity and solid boundary velocity respectively, d _s and d _f represent the solid and fluid displacements respectively, n _s and n _f are the normal unit vectors of the solid and fluid boundaries respectively.

Fig. 2.

The inlet velocity.

In the calculation, the convergence criterion root mean square is set to be 1 ×10⁻⁴. According to the stability analysis of time step, the time step is set to be 0.005 s. The period is 0.8 s, and three consecutive iterations are calculated. In order to obtain a stable convergence solution, the results of the third cycle are extracted for the final processing.

3. Results

We mainly select sections S1, S2, S3, S4, S5, S6, S7 and S8 for the analysis. Among them, sections S1, S2, S3 and S4 are located in the host tube, sections S5 and S6 are located at both ends of the bypass tube, and sections S7 and S8 are located at both ends of the microcirculation. All the above sections are selected to consider the flow characteristics.

3.1. Maximum velocity

The velocity streamline diagram of the artery zone is shown in Fig. 3. The selected moment is the peak velocity moment of the blood flow cycle (0.175 s). It could be seen that the blood flows directly into the bypass tube without flowing into the narrow area, and the peak velocity occurs at the center of the blood vessel. The velocity gradually decreases after reaching the middle of the bypass tube. At the downstream anastomosis, blood also flows along the opposite direction, and there is a recirculation area.

Fig. 3.

Velocity streamline.

Figures 4 and 5 show the maximum velocity distributions at 8 selected sections in the artery and microcirculation zones during one cycle. As shown in Fig. 4, after blood flow enters into the artery from the entrance at a specified speed, the velocity at section S1 is the highest, which is slightly higher than that at section S5 during systole period (0.175 s), and the maximum velocities at sections S1 and S5 are higher than the inlet velocity. However, the velocity at section S6 is lower than the inlet velocity. There is the lowest velocity, almost zero, at section S2. That is, no blood flows into this zone. At the same time, the velocities at sections S3 and S4 is slightly higher than that at section S2. Figure 5 shows that the velocities at S7 and S8 in the microcirculation zone are extremely low.

Fig. 4.

The maximum velocity distributions in the artery zone.

Fig. 5.

The maximum velocity distributions in the microcirculation zone.

3.2. Maximum pressure

Figures 6 and 7 show the maximum pressure distributions. It could be seen in Fig. 6 that the maximum pressure distributions at the six sections in the artery zone are almost the same, while the maximum pressure variations at sections S1, S2, and S5 are much higher than those at sections S3, S4, and S6. After blood flow enters into the artery zone, the peak pressure is about 697 Pa at section S1, and about 588 Pa at section S2. In addition, there are negative pressures at section S3, and there is a high pressure drop at the narrow ends. At the same time, the pressures at section S5 are lower than those at section S2, and the pressures at section S6 are close to 0. The pressure fluctuation ranges of these sections gradually decrease. The pressure distributions at sections S7 and S8 are almost the same. There are negative pressures in the microcirculation zone.

Fig. 6.

The maximum pressure distributions in the artery zone.

Fig. 7.

The maximum pressure distributions in the microcirculation zone.

3.3. Maximum wall shear stress

The wall shear stresses at different sections are shown in Figs 8 and 9. The wall shear stress distribution at each section is almost the same, but it could be obviously observed that the wall shear stresses at sections S1, S5 and S6 have larger variations compared with those at other sections. Among them, the peak wall shear stress (11.2 Pa) exists at section S5 during the systolic period, which is higher than that at section S1 (10.8 Pa). With the flow in the bypass tube, the peak wall shear stress of about 5.28 Pa is generated at section S6. The wall shear stresses at sections S3 and S4 are slightly higher than that at section S2, but there are still low values. As shown in Fig. 9, wall shear stresses in the microcirculation zone are also extremely low.

Fig. 8.

The maximum wall shear stress distributions in the artery zone.

Fig. 9.

The maximum wall shear stress distributions in the microcirculation zone.

Figure 10 shows the wall shear stress contour at the peak systolic time (0.175 s). The wall shear stress shows a decreasing trend in the bypass tube, and a low wall shear stress of 1.35 Pa appears at the downstream anastomosis. In the narrow central area, a wall shear stress of about 6.76 Pa is generated.

Fig. 10.

Wall shear stress contour.

4. Discussion

In this work, the velocity streamline shows that blood flows almost entirely into the graft tube when it arrives at the bifurcation between the host and graft arteries, and there is no recirculation area at the anastomosis. However, in previous studies, when the fluid reached the bifurcation, most of the fluid was directed to the graft tube, and other was shunted to the narrow direction. The blood flow velocity increased at the stenosis and it gradually decreased at the downstream of the stenosis. The exit velocity in the bypass tube showed an inclination toward the outer wall of the blood vessel. This strong inclination may produce higher wall shear stress, which is a dangerous area for endothelial cell proliferation, and the backflow strength at the anastomosis is more obvious [21,22]. This work indicates that under the action of microcirculation, all fluid flows into the graft tube, and the blood flow velocity gradually decreases in the graft tube. The maximum velocities at sections S1 and S5 are higher than the inlet velocity, but with the flow of blood in the graft tube, the velocity decreases when passing through section S6, lower than the peak inlet velocity. The flow trend in the bypass tube is consistent with previous studies. After considering microcirculation, the fluid could avoid flowing into the narrow area. As well known, the junctions of the entrance and exit of the graft tube are key positions in the hemodynamics study of artery bypass graft. Fluid flow at the positions is complex and susceptible to plaque. However, the presence of microcirculation could regulate the blood flow at the entrance of the graft tube, but has no significant effect on the blood flow at the exit of the graft tube.

Similarly, the pressure and wall shear stress are also affected when microcirculation is taken into account. There are still large pressure fluctuations at sections S1, S2, and S5, reaching the maximum pressure during the systolic period and then they gradually decrease, negative pressures exist during the diastolic period, and large pressure drops are generated at both ends of the stenosis, which is consistent with the previous study [23]. The other three sections in the artery zone are always in a low pressure state close to 0 Pa in a cardiac cycle. The pressures at section S6 are slightly higher than those at sections S3 and S4, which are almost the same. This phenomenon may be caused by the partial loss of microcirculation function to regulate blood pressure.

The effect of wall shear stress on endothelial cells plays a key role in the early development of intimal hyperplasia. Too low or too high wall shear stress will have some adverse effects on endothelial cells. When wall shear stress is too low, it may promote the secretion of more damage factors in blood vessels and induce dysfunction of endothelial cell function, resulting in atherosclerotic lesions. High wall shear stress is likely to cause plaque cap rupture and direct mechanical damage to endothelial cells [24]. There is relatively high wall shear stress and it could be seen from the wall shear stress contour in Fig. 10, which is consistent with the previous study [22].

In this study, the linear elastic theory is used to simulate the arterial wall. However, the real blood vessel has viscoelastic mechanical properties, that is, nonlinear relationship. It should be noted that the nonlinear stress-strain may lead to the change of blood flow velocity, affect the flow state of blood flow in the coronary artery, and the wall shear stress may increase significantly at the site of arterial stenosis. In addition, the nonlinear stress-strain may also induce the increase of blood flow disturbance [25].

It is suggested that the 45° suture angle may be the optimal bypass graft angle for transplantation surgery, which could reduce the damage to the arterial endothelial cells in the surrounding suture region and thus significantly reduce the risk of intimal hyperplasia [26]. In this work, the model considering microcirculation function could more reasonably simulate the hemodynamic changes in the case of the bypass graft, providing a theoretical basis for clinical decision-making, and helping clinicians to adopt surgical plans according to the specific conditions of patients when performing bypass transplantation, so as to improve the success rate of surgery. It provides an approach for early surgical risk detection of arterial diseases to reduce the risk of complications. In our future work, a nonlinear blood vessel and patient-specific models will be considered to improve the accuracy and meet clinical requirements.

5. Conclusion

In this paper, an idealized artery bypass graft model with microcirculation as the outlet boundary condition is established, and the hemodynamic characteristics are observed through numerical simulation using the finite element method. The model considering microcirculation function could simulate the physiological characteristics of blood flow more reasonably. This study provides a new insight for hemodynamic modeling, which could help clinicians better diagnose and treat arterial diseases.

Footnotes

Acknowledgements

This work is supported by the National Clinical Research Center for Cardiovascular Diseases, Fuwai Hospital, Chinese Academy of Medical Sciences (No. NCRC2020007) and CAMS Innovation Fund for Medical Sciences (No. 2017-I2M-3-003).

Conflict of interest

The authors declare that they have no competing interests.

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