Numerical Analysis of Fluid-Structure Interaction in The Aortic Arch Considering Various Blood Flow Rates
محورهای موضوعی : Mechanical Engineering
Hamid Zandvakili
1
(PhD candidate Science and Research branch, Islamic Azad University, Iran)
Kamran Hasani
2
(Associate Professor, Science and Research branch, Islamic Azad University, Iran)
Syamak Khorramymehr
3
(Science and Research branch, Islamic Azad University, Iran)
کلید واژه: Aortic Arch, FSI Method, Hemodynamics, Numerical Modeling, Perfusion,
چکیده مقاله :
Hemodynamic forces are felt by the biomechanical receptors of the arterial wall to give an appropriate response to maintain homeostasis. On the other hand, baroreceptors are a type of biomechanical receptors that are sensitive to abnormal stretch sizes. It is very important to predict the distribution of stress and strain caused by the hemodynamic field to the vessel wall in pressure-sensitive areas to evaluate the function of these receptors. In the present study, a three-dimensional (3-D) model of the aortic arch is presented. The geometry was reconstructed based on the CT images. Also, numerical analysis was performed using the fluid-structure interaction method. First, the hemodynamic field containing the pressure and velocity distribution in the blood area was obtained. Then, the deformation and stress fields in the solid domain were analyzed. The results show that the highest vertical stress occurs in the posterior supra aorta. So, the amount of this maximum vertical stress increases up to 5 kPa in some places; these points have higher tensions, and they can be susceptible to rupture and aneurysm diseases. Higher normal stress happened at the aortic root and the supra-aortic branches and reached approximately 200 kPa at Peak Systole. Also, the highest amount of strain occurs in the posterior supra aorta, reaching 0.001.
Hemodynamic forces are felt by the biomechanical receptors of the arterial wall to give an appropriate response to maintain homeostasis. On the other hand, baroreceptors are a type of biomechanical receptors that are sensitive to abnormal stretch sizes. It is very important to predict the distribution of stress and strain caused by the hemodynamic field to the vessel wall in pressure-sensitive areas to evaluate the function of these receptors. In the present study, a three-dimensional (3-D) model of the aortic arch is presented. The geometry was reconstructed based on the CT images. Also, numerical analysis was performed using the fluid-structure interaction method. First, the hemodynamic field containing the pressure and velocity distribution in the blood area was obtained. Then, the deformation and stress fields in the solid domain were analyzed. The results show that the highest vertical stress occurs in the posterior supra aorta. So, the amount of this maximum vertical stress increases up to 5 kPa in some places; these points have higher tensions, and they can be susceptible to rupture and aneurysm diseases. Higher normal stress happened at the aortic root and the supra-aortic branches and reached approximately 200 kPa at Peak Systole. Also, the highest amount of strain occurs in the posterior supra aorta, reaching 0.001.
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Int. J. Advanced Design and Manufacturing Technology, 2024, Vol. 17, No. 2, pp. 1-6
DOI: ISSN: 2252-0406 https://admt.isfahan.iau.ir
Numerical Analysis of Fluid-Structure Interaction in The Aortic Arch Considering Various Blood Flow Rates
Hamid Zandvakili, Kamran Hassani*, Siamak Khorramymehr
Department of Biomedical Engineering, College of Medical Science and Technologies, Tehran Science and Research Branch, Islamic Azad University, Tehran, Iran
E-mail: dr.zandvakili@gmail.com, k.hasani@srbiau.ac.ir, s.khorramymehr@srbiau.ac.ir
*Corresponding author
Received: 7 February 2023, Revised: 26 July 2023, Accepted: 10 April 2023
Keywords: Aortic Arch, FSI Method, Hemodynamics, Numerical Modeling, Perfusion
Biographical notes: Hamid Zandvakili received his BSc in Mechanical Engineering from Shahid Bahonar University of Kerman. He received his MSc in biomechanics from the Science and Research Branch of Islamic Azad University and is currently a PhD candidate in biomechanical and biomedical engineering at this university. Kamran Hassani received his PhD in biomedical engineering from Amirkabir University. He is currently an associate professor at the Science and Research Branch of Islamic Azad University. His field of research is mainly cardiovascular system biomechanics. Siamak Khorramymehr received his PhD in biomedical engineering from the Science and Research Branch of Islamic Azad University. He is currently an Assistant Professor in Medical Sciences and Technologies at the Science and Research Branch of Islamic Azad University. His field of research is mainly biomechanics, ergonomics, and rehabilitation engineering.
1 Introduction
Cardiovascular diseases are one of the common diseases in today's societies that can be caused by problems in the blood vessels and the behavior of the blood in the veins. In order to recognize these diseases, there must be enough information about the behavior of the blood, such as changes in blood pressure and velocity profiles, as well as changes in tension and changes in the shape of the walls of the vessels. On the other hand, the baroreflex is a homeostasis mechanism of the body that is used to keep blood pressure constant. This reflex is involuntary and is considered a quick self-regulation. The baroreceptor system plays an important role in the short-term regulation of blood pressure, which is mediated by baroreceptors. These receptors are located in the carotid arteries, the aortic arch, and the walls of the heart cavities and are activated in response to the increase in blood pressure. High blood pressure causes the heart rate to decrease, and then the blood pressure decreases. Low blood pressure reduces the activity of the baroreceptor reflex, and as a result, heart rate and blood pressure increase. Our current knowledge of identifying different components of the baroreceptor reflex is the result of numerous preliminary studies that have been carried out on humans and animals. Although the importance of baroreceptors in the control of arterial pressure under resting conditions has never been doubted, there are disagreements about how they function, considering various blood flow rates.
Donald and Edis, in their research on the isolated carotid baroreceptors of dogs, showed for the first time that the baroreflex performance curve undergoes a re-adjustment during exercise compared to the resting state without changing the sensitivity of the baroreceptors [1]. Later studies confirmed this finding in humans and showed that the amount of this re-regulation has a direct relationship with the intensity of exercise. Feng et al. and his colleagues conducted studies on the numerical modeling of aortic arch baroreceptors. They combined finite element modeling with laboratory work on the aortic arch of a rat. Numerical simulation in this study could show the functional role of axial loading in the baroreceptor region of the aortic arch. The laboratory studies that were done on the aortic arch of the mouse confirmed the validity of the numerical simulation [2-6]. Lee et al. proposed the CFD method on several carotid bifurcations and illustrated how the zones interacted with the connection among the geometric construction and flowed fluctuation as oscillatory shear parameters [7]. Also, Dong et al. performed blood flow downstream at the carotid branches, resulting from the downstream arterial impedance [8]. According to the FSI analysis of the aortic wall, the study of Suito et al. is performed on the thoracic aorta of the human vascular system. They constructed the geometry of the aorta based on CT images and concentrated on the connection of the WSS distribution in the centerline [9]. Lantz et al. carried out an FSI model of the human aorta using MRI images. They simulated the impact of arterial wall movement on WSS variation by considering the outcomes related to rigid arterial wall results [10]. In this regard, Crosetto et al. proposed similar mathematical modeling with an identical approach. The boundary conditions applied for the vessel outlets were received results in a one-dimensional simulation [11]. In the current study, a three-dimensional aortic arch model is presented as considering various blood flow rates. The mathematical modeling was considered by the FSI method to reach the hemodynamic forces applied to the arterial vessel wall and results in abnormal stresses. Also, more fluid details were analyzed concerning average peak and mean blood velocities corresponding to the ascending aorta inlet blood flow with respect to the previous literature review. Simulation models of different parts of the carotid vascular system have been presented in recent years [12]. A three-dimensional and accurate mathematical model can help to investigate the baroreflex system's performance accurately. The present study investigated a three-dimensional aortic arch model as the expected maximum stress and deformation locations, considering various blood flow rates.
2 Method and Materials
This research intends to quantitatively study the deformation of the aortic wall by creating an accurate three-dimensional model that considers the properties using the fluid-structure interaction (FSI) method. In order to simulate the arteries of the human body, the real dimensions were determined using Mimics software, which modeled the arteries obtained from the positron emission tomography-computed tomography (PET-CT) imaging method. In order to simulate the arteries of the human body, the real dimensions have been determined using Mimics software from the modeling of the arteries obtained from the positron emission tomography-computed tomography (PET-CT) imaging method. Then, the CT scan was transferred to computational fluid dynamics software for the fluid flow part, as well as finite element software for the solid part; modeling and factors such as stress, strain, and deformation were studied. For numerical simulations, the direct interaction between blood and the solid wall of the vessel is used. Strains and deformations will be desired to calculate the response of the flexible vessel wall. While considering the externally applied loads, the velocity, acceleration, and displacement vectors in the vessel domain, along with the pressure and flow fields, are calculated by solving the Equations of motion in the fluid domain. Also, the blood flow in the vessels of the human body is considered a Newtonian fluid that obeys the dynamic relationships of fluids with specific density and viscosity. Solving the governing Equations in solid-fluid interaction problems is similar to fluid problems with moving boundaries.
2.1. Governing Equations of The Fluid Region
The governing Equations of blood flow are the mass continuity and momentum conservation Equations. The law of conservation of mass is generally expressed as “Eq. (1)”.
(1)
In three-dimensional flow i= 1,2,3, ρ is density, u is velocity, and t is time. The momentum Equation in its general form for incompressible flow and constant viscosity is given by “Eq. (2)”.
(2)
In this Equation, fi is the volumetric force on the fluid flow, which can represent the forces acting on the fluid due to gravity.
2.2. Equations Governing the Solid Region of The Vessel
Solving the solid field in problems that include fluid-structure interaction is the same as solid problems, with the difference that after each step of solving the fluid field and its convergence, the forces acting on the solid must also be updated. In the problems of fluid-structure interaction, the solid medium generally exhibits non-linear behavior. The computational domain in fluid-structure interaction problems with dynamic elastic Equations, which include the momentum Equation and the equilibrium Equation, is described by the momentum “Eq. (3)” and the equilibrium “Eq. (4)”.
(3)
(4)
In these Equations, ρ is the density of the solid substance, 
represents the acceleration of the point mass of the solid, d is the displacement of the point mass, σ is the Cauchy stress tensor, ƒ is the external forces applied at time t, 
is the stresses applied to the external surface at time t, and n is the normal vector to the surface.
2.3. Fluid-Structure Interaction
In order to achieve more realistic results, the physical conditions governing the problem must be fully considered. In the analysis of problems, considering the interaction of fluid and solid, the effect of flow on the wall and the effect of the deformation of the solid wall on the movement of the fluid is considered. In this type of analysis, two separate models are prepared, and the computational domain is divided into two parts: fluid domain and solid domain. When solving these two domains, they should be coupled together and solved simultaneously. The solid model is based on the Lagrangian system. In the Lagrangian view, the motion of a single particle is followed, and its displacement is unknown. The fluid model alone is always analyzed with the Eulerian coordinate system. For fluid-solid interaction problems, a combination of Eulerian and Lagrangian perspectives should be used, which is known as the arbitrary Lagrangian-Eulerian system (ALE). In fact, the Eulerian point of view is used for fluid domain Equations, the Lagrangian point of view is used for solid Equations, and the ALE point of view is used for common boundaries of solid and fluid. In fluid-structure interaction problems, the stresses and deformations of the solid domain are calculated simultaneously with the flow parameters that are in contact with the solid domain. In other words, the changes in the shape of the solid domain are caused by fluid pressure gradients, and in a similar way, the pressures and velocities in the fluid domain are also dependent on the changes in the shape of the solid domain structure.
2.4. Computational Model-Building Steps
The geometry used in this project is from hospital images in DICOM format, which are taken in two dimensions from the body of a healthy person. After importing in the Mimics software, the complete geometry of the cardiovascular system is imaged by specifying the range of rays irradiated to the patient's body and based on the soft tissue, which includes the cardiovascular system as well as the respiratory system. By adjusting the lower and upper limits of the pixels, the images can be seen. Then, by selecting the geometry of the cardiovascular system, which includes the ventricles, atria, and aorta, as well as blood vessels and pulmonary vessels, it was extracted separately, as shown in “Fig. 1”.
2.5. Geometry and Boundary Conditions
Based on the research of Gabe et al. [13], the maximum and minimum average blood velocity in the ascending aorta is 
and 
. Current studies consider three velocities of , 
, and cm/s for blood flow simulation [14]. The blood flow at the velocity of is laminar. However, the blood flow at the velocities of and 0.385 cm/s the blood flow is turbulent. The present study is simulated with the 
turbulence model based on the study of James et al. [15]. Based on the assumption of Vinoth et al. [16], boundary condition data related to a relative pressure of 
Hg is assumed at all aortic branch outlets. Blood density is 1050 
, and its dynamic viscosity is 
. Based on Giannakoulas et al.'s study [17], Young's modulus is equal to 
, Poisson's ratio is 
, and a density of
was considered. For all inlets and outlets, the fixed support boundary condition is applied [18]. The amount of displacement in fixed support boundary conditions is zero.
Fig. 1 The generated geometry of the cardiovascular system extracted by Mimics software for a 45-year-old male.
2.6. Mesh Construction
Due to the curved geometry of the blood vessel and its asymmetry during grid generation, it has been tried to use the meshing of tetrahedral elements and boundary layer elements in the vessel wall to achieve more accurate results. The total number of meshes used is selected after checking meshing independence. The grid is made with average Orthogonal Quality=0.78, Skewness=0.21, and Aspect Ratio=1.80 (“Fig. 2”).
Fig. 2 The geometry of the aorta and the mesh created in the computational model.
In this simulation, the mesh sensitivity analysis was studied. It needs a mesh sensitivity analysis to ensure the problem is independent of the mesh size. In this research, several mesh samples with different sizes and inflation mesh on the vessel walls with a growth rate of 1.2 have been used to achieve better results. Current results considered the amount of equivalent pressure created at the vessel wall as a criterion for the appropriateness of the number of calculation elements, as shown in “Fig. 3”.
Fig. 3 Mesh sensitivity analysis in the computational model.
The measured maximum and minimum pressure at each location are shown in “Table 1”. The measured pressures are relatively low, which is probably due to the patient being anesthetized (part of the standard clinical procedure). The pressure drops, as opposed to the absolute pressure, define the flow through the vessel in the absence of interaction between the fluid and the mechanical properties of the vessel wall; thus, the relatively lower magnitude of the measured pressures would not affect the predicted flow under the present simulation conditions. Table 1 shows that the individual values of Pmax and Pmin at each outlet are within 6% of the pulse pressure. The pulse pressure at each outlet is within 3.5%. In the AA, the maximum pressure is close to the value measured and the mismatch for minimum pressure is 8% of the pulse pressure. Two additional invasive measurements in the distal arch and mid-abdominal aorta (in the TL) were compared with the CFD values for further validation. These were found to be in close agreement as shown in “Table 1” [19].
Table 1 Minimum and maximum pressure values at various locations throughout the domain, measured experimentally and derived from the simulations. The boundaries are labelled: AA–ascending aorta, BT–brachiocephalic trunk, LCC – left common carotid artery, LS–left subclavian artery, DA–descending aorta
| Pressure | AA | BT | LCC | LS | DA |
Exp. | Pmin | 56 | 50 | 50 | N/A | 52 |
Pmax | 103 | 97 | 96 | N/A | 99 | |
CFD | Pmin | 52.7 | 52 | 51.9 | 52.3 | 51.1 |
Pmax | 103.3 | 97.9 | 97.3 | 97.6 | 98.1 |
3 Results and Discussion
This research used the FSI method based on the leading models obtained from the aorta by real images obtained from PET-CT imaging devices. After running in the Mimics software for modeling based on the actual dimensions and geometry of the vessel, the obtained model is transferred to the Ansys software for analysis and is analyzed in different conditions using the finite volume method. In the present study, blood is considered a Newtonian, incompressible, and isothermal fluid, which is continuous. In simulating the geometry of the aorta, the continuity and Navier-Stokes Equation are the governing Equations for the 3-D blood movement. Also, the boundary condition of the fluid and solid interaction introduces forces from the blood side to the vessel wall. This section draws the pressure and velocity contours for the blood flow domain and the deformation, strain, and stress contours for the vessel's solid domain. These contours are drawn for three blood inflow velocities of 11 cm/s, 38.5 cm/s, and 66 cm/s. Figure 4 shows the static pressure contours in the middle and transverse planes. The maximum amount of pressure occurs in the supra-aorta, reaching 16.04 kPa, 16.41 kPa, and 17.32 kPa at velocities of 11 cm/s, 38.5 cm/s, and 66 cm/s, respectively. Baroreceptors are located at the place of application of this maximum pressure. Also, the lowest pressure occurs in the descending aorta, and the minimum amount at the velocities of 11 cm/s, 38.5 cm/s, and 66 cm/s is 15.98 kPa, 15.84 kPa, and 15.58 kPa, respectively. Therefore, in the descending aorta, the low-pressure area causes blood to be sucked toward the abdominal aorta.
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