Flow through a heart artery is modeled using computational fluid dynamics with and without a more detailed fluid-solid interaction model. The study describes the interactions between the blood flow and the arterial wall. Blood rheological properties are modeled using the Bird-Carreau model under steady state conditions. The flow path is comprised by the right coronary artery and two branches (bifurcations) namely the acute marginal and the posterior descending sections. Inlet velocity values of a typical cardiac cycle are used in the model. The differences between the solutions from the FSI model and the CFD model were determined by comparing the wall shear stress and the first principal stress. Wall shear stresses and first principal stresses for four separate times (0.05, 0.14, 0.44 and 0.96 s) are reported. At 0.14 s and at 0.96 s, the values of the WSS maximum are greater than those previously computed using FSI by 10.8 Pa and 7.5 Pa. Similarly at 0.44 s, significant differences were obtained in the WSS distributions between both solutions. The highest magnitudes of first principal stresses were of 552 kPa in the bifurcation of the PDA at a time of 0.44 s.
Research Article
Computational Simulation of Blood Flow in the Right Coronary Artery and its Interaction Between the Blood Flow and the Arterial Wall
https://doi.org/10.21203/rs.3.rs-3558194/v1
This work is licensed under a CC BY 4.0 License
Journal Publication
published 29 Mar, 2024
You are reading this latest preprint version
modeling heart pressure
arterial flow behavior
fluid solid interaction
finite element methods
artery wall stress
In the last decade, different computational models have been published to simulate blood hemodynamic phenomena in coronary arteries by means of Computational Fluid Dynamics (CFD) and Fluid-Solid Interaction (FSI) simulations [1]. Considering only the fluid, CFD models are less computationally demanding and are used to simulate these phenomena. FSI models describe the coupling between arterial wall mechanics and the blood flow. Consequently, the complexity and the difficulty of convergence of these models increase [2].
The development of FSI computational models for simulating the fully coupled interaction between the blood flow and the wall of a coronary artery implies handling the geometrical representation of the domains, the blood flow profile, the rheology, the representation of the arterial wall and the validation of the results.
The geometry of the coronary artery has been considered as a simple pipeline of constant diameter which implies a significant simplification of the problem [3, 4] and facilitates the convergence of various tests performed with different rheological models [3, 4]. More complex geometries are generated by means of 3D reconstructions of CT (computer tomography), CTA (computed tomography angiography) [5], IVUS (intravascular ultrasound) [6], MRI (magnetic resonance imaging), angiogram images, and software, such as MIMICS® (Materialise, Ann Arbor, MI, USA) [2, 7], SimVascular (open-source)[8, 9] and non-commercial open-source reconstruction software [10, 11].
Blood flow in a healthy right coronary artery (RCA) has a Reynolds number of about 200 in resting conditions [12]. The corresponding laminar profile is distorted into a jet flow profile when the RCA lumen decreases in diameter [13, 14]. Under these new conditions of higher speeds in the RCA some authors have resorted to the Large Eddy Simulation (LES)[4] and Reynolds Averaged Navier Stokes (RANS) equations [13]. Blood has been modeled as a Newtonian fluid because under high shear and large lumen diameter like the RCA, the viscosity of the blood is approximately constant [15]. More detailed approaches model blood as a non-Newtonian shear-thinning fluid with rheological models such as the Bird-Carreau model [2, 3, 15, 16], Power Law model[10] and Generalized Power Law model [11, 17].
Modeling RCA arterial wall mechanical behavior is described as an elastic isotropic [7, 18] or an isotropic highly [19] hyperelastic material [2, 9, 10]. The Mooney-Rivlin equations have been used to represent the hyperelastic behavior of the RCA arterial wall with satisfactory results [2, 10].
Comparison of computed and measured hemodynamic parameters, like the wall shear stress (WSS), is the preferred route to validate the computational simulation of blood flow through the coronary artery [10, 17, 20–22]. The WSS is also the base parameter with which other hemodynamic descriptors such as the Time Average Wall Shear Stress (TAWSS), the Oscillatory Shear Index (OSI) and the Relative Residence Time (RRT) are calculated [2]. The experimental determination of these parameters presents significant challenges because determining local velocity and pressure profiles within the artery require access to the patients [2–5, 7–9, 11, 13, 16–18, 20–30].
Chaichana et al. [31] quantified the effects on WSS generated by plaque formation on coronary arterial walls with a CFD computational analysis. It was noted that plaque usually is located in the bifurcated regions [32, 33]. High WSS are located in the zones of greatest reduction of the lumen diameter, while the low WSS zones are located in the zones before and after the reduction [31].
Li et al. [21] indicate that the low WSS regions (≤ 4 Pa) occurred at the inner wall opposite to the flow divider while high WSS (≥ 40 Pa) were located on the opposite side. Pinto et al. [3] reported WSS values up to 45 Pa for RCA located near of the flow separation (bifurcation) region.
Gholipour et al. [10] calculated the time dependent variation of WSS at the location of maximum WSS for a complete cardiac cycle. They reported a range of values between 6 to 50 Pa for an RCA, without flow separation for a patient that presented ST-Elevation Myocardial Infarction.
Claes et al. [34] reported circumferential tensile strength values of coronary wall specimens of 1.5 MPa for 20 years old patients and up to 0.5 MPa for patients older than 30 years old. Liu et al. [24] simulated the von Mises stresses (VMS) of the RCA of 12 patients, in one case, they found von Mises stress magnitudes of up to 95 kPa in the main branch of this artery and up to 110 kPa in the areas near the flow separations (bifurcations). Similarly, Liu et al. [24] concluded that baseline low VMS was associated with decrease in plaque volume and decrease in minimal lumen area whereas baseline high VMS was associated with increase in plaque volume and MLA. Dong et al. [35] used the first principal stress for stress distribution analysis in the bifurcation of a coronary artery. This article showed the maximum value of first principal stresses occurred at the bifurcation vertex, followed by the bifurcation shoulders. The maximum value of first principal stress, located at the angulation, was found to negatively correlate with magnitude of the bifurcation angle.
Two simulations were developed, one that considered only the fluid and the second that accounted for interactions between fluid and the wall. The simulations were carried out in a High-Performance type virtual machine “HC44-32rs” with 32 vCPUs, 352 GB of RAM and 700 GB of mechanical disk storage.
2.1. Model Parameters
Blood was modeled as a homogeneous incompressible non-Newtonian fluid. The flow regime is laminar within the artery. They dynamic behavior of the fluid and the wall is coupled. The temperature was constant at 36.2°C, blood density and artery wall density were set as 1060 and 1120 kg/m3, respectively. The effects of gravity were neglected.
2.2. Coronary Artery Topology
A segment of the right coronary artery with the acute marginal (AM) and the posterior descendant (PDA) branches was generated. The dimensions for the segments are the following [36]:
| Dimension | Magnitude (mm) |
|---|---|
| Proximal main branch (Lumen Diameter) | 4.21 ± 0.64 |
| Middle main branch (Lumen Diameter) | 3.36 ± 0.57 |
| Distal main branch (Lumen Diameter) | 2.37 ± 0.53 |
| Acute Marginal Artery (Lumen Diameter) | 1.6 ± 0.42 mm |
| Posterior Descending Artery (Lumen Diameter) | 2.09 ± 0.62 mm |
| Artery wall thickness | 0.75 mm |
2.3. Conservation and constitutive equations
The mass and momentum conservation (Navier-Stokes) equations are the following:
$$\rho \nabla \bullet \left(\varvec{u}\right)=0$$
1
$$\rho \left(\frac{\partial \varvec{u}}{\partial t}+\varvec{u}\bullet \nabla u\right)=-\nabla p+\nabla \bullet \left(\mu \left(\nabla \varvec{u}+{\left(\nabla \varvec{u}\right)}^{T}\right)\right)+\varvec{F}$$
2
where \(\rho\) is the density, \(\varvec{u}\) is the velocity vector, \(\mu\) is the dynamic viscosity, \(p\) is the pressure, and \(\varvec{F}\) are the external volumetric forces that can affect the fluid [37]. In the case of the blood viscosity, due to its non-Newtonian behavior, we used the Bird-Carreau viscosity model, given by Eq. (3).
$$\mu ={\mu }_{\infty }+\left({\mu }_{0}-{\mu }_{\infty }\right){\left[1+{\left(\lambda \dot{\gamma }\right)}^{2}\right]}^{\frac{n-1}{2}}$$
3
where \(\mu\) is the apparent viscosity of the fluid, \(\dot{\gamma }\) is the shear rate of the blood, the other parameters have the following values:
| Parameter | Value | Units |
|---|---|---|
| \({\mu }_{0}\) | 0.056 | Pa·s |
| \({\mu }_{{\infty }}\) | 0.00345 | Pa·s |
| \(\lambda\) | 3.313 | dimensionless |
Modeling of the solid mechanics is computed via Newton's second law Eq. (4)):
$$\rho \frac{{\partial }^{2}\varvec{u}}{\partial {t}^{2}}=\nabla \bullet {\left(\varvec{F} \varvec{S}\right)}^{T}, \varvec{F}=\varvec{I}+\nabla \varvec{u}$$
4
where \(\rho\) is the density of the arterial wall, \(\varvec{F}\) is the strain gradient tensor, \(\varvec{u}\) is the displacement of the material and \(\varvec{S}\) is the second Piola Kirchhoff stress tensor. For hyperelastic materials \(\varvec{S}\) is defined as follows [37]:
$$\varvec{S}={\varvec{S}}_{ex}+\frac{\partial {\varvec{W}}_{s}}{\partial \epsilon }$$
5
where \({S}_{ex}\) is the extra stress caused by possible initial stress contributions and viscoelastic stresses, \({W}_{s}\) represents the strain energy-density function and \(\epsilon\) is the total deformation.
The arterial wall is a hyperelastic material where \(\varvec{\epsilon }\) is defined as follows [37]:
$$\varvec{\epsilon }=\frac{1}{2}\left({\varvec{F}}^{T}\varvec{F}-\varvec{I}\right)$$
6
\(\varvec{\epsilon }\) is the total deformation and \(\varvec{F}\) is the strain gradient tensor. \({W}_{s}\) was defined using the Mooney-Rivlin model according to:
$$W={a}_{10}\left({I}_{1}-3\right)+{a}_{01}\left({I}_{2}-3\right)+{a}_{20}{\left({I}_{1}-3\right)}^{2}+{a}_{11}\left({I}_{1}-3\right)\left({I}_{2}-3\right)+\dots$$
$$+{a}_{02}{\left({I}_{2}-3\right)}^{2}+\frac{1}{d}{(J-1)}^{2}$$
6
where \({I}_{1}\) y \({I}_{2}\) are the first and second strain invariants and \(J\) is the elastic volume ratio. The equation parameters have the following values:
| Parameter | Value |
|---|---|
| \({a}_{10}\) | -4.02 |
| \({a}_{01}\) | 4.321 |
| \({a}_{20}\) | 18.401 |
| \({a}_{02}\) | 39.105 |
| \({a}_{11}\) | -51.856 |
| \(d\) | 0.822 |
2.4. Boundary conditions
2.4.1 Blood flow domain
The inflow velocity was described by a time dependent function shown in Fig. 2. At arterial exits time-dependent pressures were defined as depicted in Fig. 3. Both functions were obtained for a patient with unobstructed coronary arteries and reported by Liu et al. [21].
No-slip conditions are applied on arterial walls. The artery wall is flexible and in contact with the fluid.
2.4.2 Artery wall description
The inflow and outflow surfaces of the arterial wall were fixed in place. The rest of the artery was free to move. The motion of the heart as a whole is not integrated into the model.
2.5 Domain discretization: Meshing
The mesh is divided into two domains, one is the fluid, and the other is the artery wall. The fluid mesh is a deformable mesh with hyperelastic type smoothing. The center line of the artery was defined as the reference of the displacement of the mesh.
To observe the behavior of the solution as a function of the mesh size, three computational meshes were generated with 71 029, 453 614 and 2 077 328 finite elements. The fluid and solid interactions are computed in two sets.
The first set of computations was comprised by a time dependent study of only the fluid, using the complete cardiac cycle defined by Figs. 2 and 3, with a step of 0.001 s and with a range from 0 s to 1 s.
The second set of computations included the fluid and the wall. The study determined the profiles of both the arterial wall and fluid. A stationary study was performed first to determine suitable initial conditions for the time dependent computations. The time dependent computations were performed using an auxiliary sweep at a step of 0.001 s and with a range from 0 to 1 s.
A High-Performance Azure virtual machine HC44-32rs was used. This virtual machine had 32 vCPUs, 352 GB of RAM and 700 GB of physical disk storage [38]
Wall shear stress (WSS) and first principal stresses were determined and compared with those published by [21, 22, 34, 35].
3.1 Model solution
Eight points were defined on the surface of the blood flow domain and the WSS values were evaluated at those points (Fig. 4).
|
No. Elements |
WSS (Pa) Point 1 |
Error (%) |
WSS (Pa) Point 2 |
Error (%) |
WSS (Pa) Point 3 |
Error (%) |
WSS (Pa) Point 4 |
Error (%) |
|---|---|---|---|---|---|---|---|---|
|
71k |
9.25 |
5.09 |
27.20 |
1.55 |
||||
|
453k |
8.95 |
3.35 |
3.89 |
30.92 |
26.68 |
1.96 |
0.77 |
100.57 |
|
2077k |
9.93 |
9.86 |
3.36 |
15.62 |
26.66 |
0.06 |
0.78 |
1.53 |
Tables 4 and 5 give the WSS values obtained with respect to the number of finite elements used in the simulation. As the number of finite elements increases, the approximate relative error associated with the WSS evaluation tends generally to decrease, with the exception of point 1. The reduction of the error varies at each point, which indicates that there are areas of the artery where the convergence of WSS is faster compared to other. The point that shows the greatest convergence is point 3, located in the AM close to the bifurcation, with a relative error 0.06%. Using three computational meshes with different number of elements, the highest relative error occurs at Point 4 with 453k finite elements, and a value of 100.57%. This relative error is reduced to 1.53% when using the 2077k finite elements. The greatest difference in magnitude of the WSS between the three different meshes is less than 1 Pa.
3.2 Wall Shear Stress (WSS)
The WSS determined using CFD and FSI provide a measure that may help to identify areas that in a short term might develop plaque formation in the RCA [21]. Low WSS values of up to 4 Pa and high WSS values greater than 40 Pa have been reported for simulated RCA with their bifurcations [21]. In the case of simulations of healthy RCA without its bifurcations, WSS have been obtained in the range of 0 to 8 Pa [22]. Also, for RCA with a reduction in lumen diameter as a result of plaque propagation, WSS of up to 52 Pa have been reported [10].
At 0.05 s, the simulated cardiac cycle is initiating the systole stage and the blood flow velocity reaches up to 0.33 m/s. The WSS are of up to 8.37 Pa, with a maximum of 3 Pa in the proximal and middle segments of the RCA’s main branch with a homogeneous WSS distribution. This is expected because the geometry of the sections of the RCA’s main branch do not present elements that interfere with the blood flow. The opposite occurs with WSS stresses with magnitude greater than 4 Pa in the areas close to the AM and PDA bifurcations and the distal section of the main branch of the RCA. In these areas, there are significant changes in wall stresses caused by the bifurcation’s geometry.
Figure 6 shows a maximum WSS value of 8 Pa in the inner side of the AM bifurcation curvature, in agreement with the results by Liu et al. [21]
At 0.14 s and 0.96 s the highest WSS magnitudes are generated with maximum WSS values of 85 and 50 Pa, respectively. This range of maximum WSS values is in agreement with the values (4 to 52 Pa) obtained by Gholipour et al [10]. Torii et al [22] simulated an RCA without its branches and obtained WSS values of up to 8 Pa at peak velocity time. As seen in Fig. 5b, at 0.14 s WSS values of up to 12 Pa were obtained in the proximal and media segment of the main RCA. The high WSS values obtained by the FSI at 0.14 s (peak velocity) when compared with the WSS by [22] shows the importance of taking into account RCA branches.
For all four time instances studied, the maximum value of WSS in the locations far from the two bifurcations ranged from 8.5x10− 3 to 30 Pa.
3.3 Effects of the Fluid Solid Interaction (FSI) model over the Wall Shear Stress (WSS)
In the case of a model using fluid-solid interaction, the simulated geometry includes both in the fluid and solid domains that are fully coupled. This is an improvement from using only the fluid flow description and attempting to represent the presence of a flexible wall using boundary conditions. The computational results give arterial wall displacements of up to 0.59 mm, modifying the dimensions of the lumen diameter of the coronary artery and, consequently, the magnitude of the blood flow velocity. The magnitude of the WSS is affected by these geometric changes.
Figure 7 shows the WSS magnitudes obtained simulated using only CFD. At 0.14 s and 0.96 s, the values of the WSS maximum are greater than those previously computed using FSI by 10.8 Pa and 7.5 Pa, respectively. Contrary, at 0.05 s and 0.44 s, WSS maximum values were lesser than those computed using FSI by 0.7 Pa and 0.49 Pa, respectively. At these last two times, the difference between the maximum values obtained by the FSI and the only CFD models is less than 1 Pa. But, at 0.44 s there is a significant difference in the distribution of the WSS at the middle section of the main branch RCA. The stress distribution on the main branch of the RCA is less homogeneous in the CFD-only solution than in the FSI solution. Stress zones of 4 Pa are observed along the main branch of the RCA until reaching the PDA bifurcation. In this last zone, stresses of up to 6 Pa are shown, see Fig. 7C. At 0.14 s and 0.96 s the blood flow is in a high-velocity low-pressure state, at 0.05 in a low-velocity low-pressure state and at 0.44 in a low-velocity high-pressure state. At 0.44 s, the arterial wall presented the greatest increase in lumen diameter as a result of the deformation of the arterial wall. The FSI model adds the effect to the WSS distribution, a feature that the fixed mesh of the CFD-only model does not include and is important, specially at times when blood flow is at a high-pressure state. At times of high-velocity and low-pressure, the difference between the FSI model and the CFD-only model is a magnitude of 10.8 Pa of the maximum WSS.
3.4 First Principal Stresses
The first principal stresses at 0.05 s show a maximum stress of 262 kPa (Fig. 8A). This maximum stress is on the intersection between the main branch of the RCA and the PDA, more exactly in the middle zone of the bifurcation, which has a "Y" shape. This region of high stresses is due to the RCA geometry with an acute angle that concentrates stresses.
Figure 8B depicts the arterial wall of the bifurcation between the RCA and the AM. In this bifurcation there are high first principal stresses, with values of up to140 kPa. Opposite to this zone of the arterial wall, there is another region with high first principal stress close to 85 kPa. The arterial wall of the bifurcation has also two opposite areas of low stresses, less than 20 kPa and up to -9.62 kPa. Thus, there are two opposite sites with high first principal stress and two opposite sites with low first principal stress, as shown in Fig. 8B.
This pattern of first principal stress distribution also occurs at the PDA bifurcation, as shown in Fig. 8A. On the other hand, two low stress regions of less than 20 kPa are found at opposite positions with each other, as marked by the black arrows. A high first principal stress value of 262 kPa, as discussed before, is located between these two regions.
For both bifurcations, the higher first principal stress areas are consequence of the small angle of the bifurcations. Dong et al [35] simulated left coronary artery and its circumflex artery bifurcation and found that the bifurcation inner angle negatively correlates with the first principal stress value at the bifurcation apex. Considering that, the internal angle of the PDA bifurcation is more acute than the AM bifurcation. The maximum stress values were obtained at apex of the PDA bifurcation (140 kPa for the AM and 256 kPa for the PDA) as shown in Fig. 8.
Figure 9 shows the first principal stresses obtained using FSI at 0.05 s, 0.14 s, 0.44 s and 0.96 s. At 0.44 s, there are the highest magnitudes of the first principal stresses, reaching values of approximately 110 kPa in the main branch of the artery, 350 kPa in the AM bifurcation and 571 kPa in the PDA bifurcation. The higher and lower stress areas remain at the same positions in this study. Dong et al[35] obtained principal first stress values for the bifurcation of the circumflex artery and the left coronary artery, of up to 400 kPa in the area of inner angulation of the bifurcation and of up to 270 kPa for the area at the outer angulation at peak systole (pressure) time. The values obtained in this study at 0.44 (peak systole) of 350 kPa (AM branch) at the apex of the bifurcation are in accordance with those values (400 kPa at the apex of the bifurcation)[35], considering the geometry differences between the RCA and the LCA.
Regarding the circumferential resistance to tension of the RCA’s arterial wall, Claes et al. [34] obtained circumferential tensile strength values of approximately 1.5 MPa for patients aged 20 years, and up to 0.5 MPa for patients older than 30 years. Our values of first principal stresses for the main branch of the RCA are smaller than those reported by Claes et al. [2]. Finally, values greater than 0.5 MPa are obtained around the PDA bifurcation, for the time instance of 0.44 s.
3.5 Additional considerations
Only an RCA geometry with two of its branches was generated for the study due to the lack of patient-specific data and extremely time-consuming model 3D reconstruction procedure.
In addition, the displacement of the arterial wall produced by cardiac movements was completely ignored. So, the model ignores the influence of cardiac contractions on the artery.
The solution of the 2077k finite element FSI model was obtained by means of cloud computing. The cloud service offered three variants of this virtual machine, differing in the number of vCPUs they had (16 vCPUs, 32 vCPUs and 44 vCPUs). We used the virtual machine with 32 vCPU because of a better cost/performance balance [38].
This article compared a base CFD modeling of a coronary artery with the results obtain using CFD complemented by a FSI model. In this FSI model, a representative geometry of the RCA was considered: a fluid domain (blood) and a hyperelastic domain (arterial wall). To better observe the effects of the solid-fluid interaction, the simulations were carried out in two stages. The first, consisted only of the fluid domain simulations and the second consisted of the simulations of solid-and-fluid domains with the addition of a FSI model.
The WSS obtained in the FSI solution and CFD-only present significant differences between them. At 0.14 s and at 0.96 s, the values of the WSS maximum are greater than those previously computed using FSI by 10.8 Pa and 7.5 Pa. Similarly at 0.44 s, significant differences were obtained in the WSS distributions between both solutions. The WSS distribution of the FSI solution is more homogeneous than that obtained in the CFD-only solution, evidencing the effect of the deformation of the arterial wall product of the blood flow on the hemodynamics inside the RCA. The magnitude of the first principal stresses obtained at four different times (0.05 s, 0.14 s, 0.44 s and 0.96 s) are below the circumferential resistance to arterial wall pressure of the RCA of 1.5 MPa for 20-year-old patients [34]. The highest magnitudes of first principal stresses were of 552 kPa in the bifurcation of the PDA at a time of 0.44 s.
Author contributions
All authors contributed to the study conception and design. Parameter collection, computational model simulation and analysis were performed by Manuel A. Molina Campos, Benito A. Stradi Granados, and Bruno Chiné. The first draft of the manuscript was written by Manuel A. Molina Campos and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Conflicts of interests
The authors of this paper confirm that there is no any conflict related to the content of this manuscript.
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