Blood Flow Simulation in Bifurcating Arteries: A Multiscale Approach After Fenestrated and Branched Endovascular Aneurysm Repair

Abstract

Pathophysiological conditions in arteries, such as stenosis or aneurysms, have a great impact on blood flow dynamics enforcing the numerical study of such pathologies. Computational fluid dynamics (CFD) could provide the means for the calculation and interpretation of pressure and velocity fields, wall stresses, and important biomedical factors in such pathologies. Additionally, most of these pathological conditions are connected with geometric vessel changes. In this study, the numerical solution of the 2D flow in a branching artery and a multiscale model of 3D flow are presented utilizing CFD. In the 3D case, a multiscale approach (3D and 0D–1D) is pursued, in which a dynamically altered velocity parabolic profile is applied at the inlet of the geometry. The obtained waveforms are derived from a 0D–1D mathematical model of the entire arterial tree. The geometries of interest are patient-specific 3D reconstructed abdominal aortic aneurysms after fenestrated (FEVAR) and branched endovascular aneurysm repair (BEVAR). Critical hemodynamic parameters such as velocity, wall shear stress, time averaged wall shear stress, and local normalized helicity are presented, evaluated, and compared.

1. Introduction

Pathological conditions in arteries are crucial for the health of the patient. Common pathological conditions that can be observed are stenotic arteries and aneurysms. The simulations utilizing Computational fluid dynamics (CFD) are the most reliable solution for evaluating the patient’s condition through flow parameters such as velocity, pressure, stresses at the wall, and more advanced biomedical parameters.

In the cardiovascular system, bifurcation regions are present, with the abdominal aortic bifurcation being of great interest, as it is the largest bifurcated artery in our body. Fluid mechanics contribute to flow analysis and provide crucial information that must be assessed individually for each case. Geometric parameters of the artery, hemodynamic factors, and boundary pulsatility significantly influence the stenosis progression and its narrowing, leading to pathological conditions for the patient.

Branching networks are essential in the human body and have been extensively studied. Pedlay et al. have examined such networks of branching tubes with varying geometry [1]. Pradham and Guha presented a 3D parametric study in a bifurcation [2]. Wilde et al. studied several shear stress factors in the carotid bifurcation of mouse-specific FSI (Fluid Structure Interaction) cases [3]. Furthermore, Lewandowska et al. conducted a comparative study on different carotid bifurcation angles [4]. Additionally, Ahmed and Podder explore how diseases, such as anemia and diabetes, influence fluid dynamic parameters [5]. Finally, Garcia et al. analyze local hemodynamic changes in a coronary bifurcation caused by three different stenting techniques [6], while Younis et al. apply an FSI approach based on the finite element method to investigate hemodynamic flow parameters [7].

The evolution of vortices and secondary motion including Dean and anti-Dean type structures is analyzed in [8]. The flow dynamics in multiple generations, with multiple branches, are investigated by many researchers [9,10,11,12] and are of great interest as they represent the complex structure of many organs of the human body. Blood flow in the normal cardiovascular system is described as laminar but, on several occasions, could operate to the threshold of turbulence. Morphological distortions, such as vascular stenosis, could cause a transition to turbulent blood flow. Specifically, diseases such as aortic valve stenosis and aortic coarctation are associated with turbulent flow in the aorta. High velocities and low blood viscosity, as occurs in anemia due to the reduction of hematocrit, are more likely to cause turbulence. An increase in diameter without a change in velocity also increases the Reynolds number. Under ideal conditions—e.g., long, straight, smooth blood vessels—the critical Reynolds number is relatively high. However, in branching vessels or vessels with atherosclerotic plaques protruding into the lumen or aneurysm, the critical Reynolds number is much lower, so turbulence could easily occur even at normal flow velocities.

In addition, mathematical modeling of blood flow in systemic arteries is of great interest due to its significance in human health. In this study, we employ an advanced mathematical model for monitoring pressure and flow profiles at specific locations, as these profiles are vital for the final behavior of blood flow in the abdominal aorta. This 0D–1D model predicts the flow and pressure at any given time in the systemic arteries of interest. Blood is treated as an incompressible Newtonian fluid, while vessels are assumed to be elastic. To ensure a reliable numerical solution, the model is divided into the first part (1D), which includes the large arteries, and the second part, which includes the smaller ones (0D) [13]. For the smaller arteries, a Windkessel mathematical model is used as described in [13]. For larger arteries, Navier–Stokes, continuity, and arterial wall motion are solved as a coupled system of equations. The solution of the system of equations (0D–1D) for the entire arterial tree provides the boundary conditions that are applied in the 3D problem that is described by the three-dimensional Navier–Stokes equations.

Endovascular aneurysm repair (EVAR) is a reliable alternative to open surgery, but commercial endografts are not dedicated to all thoracic and abdominal aortic anatomies. The scope of this study is to compare the properties of blood flow in the splanchnic (secondary) vessels after F/BEVAR stent designs.

Fenestrated devices are made with reference to the patient’s unique anatomy. Fenestrated grafts include small openings that allow the restoration of blood flow to the arteries branching from the aorta, which deliver blood to essential organs such as the kidneys, bowel, and liver. Fenestrated stent grafts use the healthier part of the abdominal aorta as a landing zone and are deployed under fluoroscopic guidance. Their fenestrations are connected to the target vessels with flared stent grafts for secure blood flow.

Branched devices are endografts that have specific branches for the main target vessels. The branches have a downward direction and are useful for the treatment of complex, mostly thoracoabdominal, aneurysms. They are used in the case of suprarenal aneurysms with or without angulation.

The selection of the stent device depends on the unique anatomy of the patient’s aorta, and until now, there has not been an optimal combination of B/FEVAR. The impact of each device on blood flow, influenced by the respective stent, must be considered when selecting between the two options.

The study of hemodynamic parameters and the significant changes of different endograft designs have been studied through CFD [13,14,15]. The visceral arteries may play a vital role in the mathematical modeling and could create small flow perturbations that generally affect the patient’s health. Changes in important hemodynamic parameters between preoperative and postoperative FEVAR cases have been evaluated via CFD, resulting in all hemodynamic characteristics converging to normal after repair [13,14,15,16]. Regarding the effectiveness of BEVAR, many studies have been conducted on their postoperative impact and the reasons for their long-term effects [14,16,17,18,19,20].

In this study, two-dimensional (2D) bifurcation flow and 3D patient-specific numerical solutions are presented. For the two-dimensional case, the solution is acquired via the use of the Finite Volume Method and a Newton-like algorithm. Levenberg–Marquardt is utilized to solve the non-linear system of PDEs (Navier–Stokes and continuity equations), aiming to obtain robust and converged numerical solutions. The main advantage of this method is its reliable solutions to stiff problems, where classic iterative methods may fail to deliver. The pressure and velocity fields are presented in both bifurcated and stenotic bifurcated geometries. The impact of stenosis near a branching network is highlighted through the contours of the pressure distribution.

Regarding the multiscale 3D study, the main focus was on the hemodynamic changes in F/BEVAR cases affecting the four visceral (secondary) target vessels. The results are exported using an advanced multiscale mathematical model introduced in [13,15]. The two different cases of B/FEVAR are simulated, compared, and evaluated. The hemodynamic parameters of interest are exported through CFD simulations. The most significant of these, including wall shear stress (WSS), localized normal helicity (LNH), and time averaged WSS (TAWSS), are presented in this study.

2. Mathematical Formulation

2.1. The 2D Study

The mathematical modeling of the problem is described by the Navier–Stokes and continuity partial differential equations (PDEs) for incompressible flow. The momentum equations of interest are provided in vector form,

$${\displaystyle \frac{\partial \overrightarrow{q}}{\partial t}}+(\overrightarrow{q}·\overrightarrow{\nabla })\overrightarrow{q}=-{\displaystyle \frac{1}{\rho }}\overrightarrow{\nabla }p+\nu {\overrightarrow{\nabla }}^2\overrightarrow{q},$$

(1)

where p is the pressure, $\rho$ the density, and $\nu$ is the kinematic viscosity. The equations that describe the two-dimensional, steady-state, and incompressible flow are the system of PDEs that includes x, y momentum, and the continuity equation. The system of PDEs is provided below in dimensionless form,

Continuity equation, dimensionless

$${\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}=0,$$

(2)

x-momentum equation, dimensionless

$${\displaystyle \frac{\partial \left(u^2\right)}{\partial x}}+{\displaystyle \frac{\partial \left(uv\right)}{\partial y}}=-{\displaystyle \frac{\partial p}{\partial x}}+{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \frac{1}{Re}}{\displaystyle \frac{\partial u}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle \frac{1}{Re}}{\displaystyle \frac{\partial u}{\partial y}}\right),$$

(3)

y-momentum equation, dimensionless

$${\displaystyle \frac{\partial \left(uv\right)}{\partial x}}+{\displaystyle \frac{\partial \left(v^2\right)}{\partial y}}=-{\displaystyle \frac{\partial p}{\partial y}}+{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \frac{1}{Re}}{\displaystyle \frac{\partial v}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle \frac{1}{Re}}{\displaystyle \frac{\partial v}{\partial y}}\right),$$

(4)

where u and v are the components of the velocity vector, $\overline{q}=(u,v)$, p is the pressure, and the Reynolds number is defined as $Re={\displaystyle \frac{\rho u_{0}L}{\mu }}$, where L is the characteristic length and $u_0$ is the maximum value of the parabolic velocity applied at the inlet.

Numerical solution. The studied problem is discretized under the Finite Volume Method, a second-order discretization method. The acquired algebraic system is strongly non-linear, and the domain of interest is divided into a finite number of control volumes. By integrating the governing equations over every control volume, the final algebraic system of equations is obtained. The system of algebraic equations for the two-dimensional case consists of three dimensionless equations for each control volume, the two momentum equations in addition to the continuity equation that are provided below.

Continuity equation, discretized

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\Delta y}{2}}{u}_E-{\displaystyle \frac{\Delta y}{2}}{u}_W+{\displaystyle \frac{\Delta x}{2}}{v}_N-{\displaystyle \frac{\Delta x}{2}}{v}_S=0,\hfill \end{array}$$

(5)

x-momentum equation, discretized

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}\Delta y\left(u_E^2-u_W^2\right)+{\displaystyle \frac{1}{2}}\Delta x\left(u_N{v}_N-u_S{v}_S\right)+\Delta y\left(p_E-p_P\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{Re}}\left({\displaystyle \frac{\Delta y}{\Delta x}}(u_E-2u_P+u_W)+{\displaystyle \frac{\Delta x}{\Delta y}}(u_N-2u_P+u_S)\right)=0,\hfill \end{array}$$

(6)

y-momentum equation, discretized

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}\Delta y\left(u_E{v}_E-u_W{v}_W\right)+{\displaystyle \frac{1}{2}}\Delta x\left(v_N^2-v_S^2\right)+\Delta x\left(p_N-p_P\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{Re}}\left({\displaystyle \frac{\Delta y}{\Delta x}}(v_E-2v_P+v_W)+{\displaystyle \frac{\Delta x}{\Delta y}}(v_N-2v_P+v_S)\right)=0.\hfill \end{array}$$

(7)

where $\Delta x$ and $\Delta y$ are the dimensions of the control volume and $E,W,P,S$ and N are the east, west, centroid, south, and north points of each control volume.

For the final numerical solution, which contains the $u,v$ and p values in every volume of the computational domain, we use a Newton-like method, specifically the Levenberg–Marquardt algorithm. Newton-like methods use the Jacobian matrix of residual functions $f_1,f_2,f_3$ that corresponds to the Equations (5)–(7) equal to zero. The derivatives of the functions are obtained with respect to the known variables $u,v,p$. The calculation of the Jacobian matrix can be computationally expensive, and the solutions that we propose lie in the sparsity of this matrix. With this solution, the algorithm applies only to the non-zero elements of the matrix. This is a common technique in Newton-like methods, as the number of zeros is exponentially increasing as the number of grid points is getting larger. A large-scale problem, such as the one we are approaching, needs to be adjusted, aiming to be optimal regarding computational memory and time consumption.

The main difference of this approach compared to other common CFD solvers such as PISO and variations of SIMPLE is that we are working in a collocated grid in which we are solving the raw Navier–Stokes equations coupled, without any change or any addition of equations like pressure correction equations. The numerical convergence criterion has been set to a very small number $\epsilon ={10}^{-15}$.

The Levenberg–Marquardt algorithm is chosen for this problem due to the robustness that it provides. The mathematical scope of this method lies in the Newton-like methods, which use trust-region frameworks. For spherical trust regions, the subproblems to be solved are described in Equation (8).

$$\underset{\overline{p}}{min}{\displaystyle \frac{1}{2}}\parallel J_k\overline{p}+{\overline{r}}_k{\parallel }^2,\mathrm{for}\parallel \overline{p}\parallel \le {\Delta }_k,$$

(8)

where $\overline{p}$ is the step direction in the parameter space, which we are trying to determine, $J_k$ is the Jacobian matrix of the residual function evaluated at the current iteration k, and ${\overline{r}}_k$ is the residual vector at iteration k. The ${\Delta }_k$ is defined as the radius of trust region, and $\parallel ·\parallel$ is the Euclidean norm. Thus, the trust-region approach requires us to solve a sequence of subproblems [21]. Finally, we obtain the optimal field of parameter $\overline{p}$, which in our problem corresponds to the velocity components $u,v$ and pressure p.

2.2. The 3D Study

The problem under consideration is governed by the unsteady and incompressible Navier–Stokes, Equation (1), and the continuity equation.

The 0D–1D mathematical model that was developed predicts blood flow and pressure in all systemic arteries of the arterial tree, and the 3D Navier–Stokes numerical solution is obtained to evaluate the post-endovascular F/BEVAR cases. For this study, post-endovascular aneurysms are constructed from computed tomography scans of patients treated with custom-made fenestrated and branched endografts. The cases presented are from patients who suffered from AAA and had a patient-specific stent graft system implanted at the General University Hospital of Larissa, Thessaly, Greece. Patients received custom-made fenestrated devices, with configurations including fenestrations for renal arteries, scallops for the superior mesenteric artery (SMA), and in some cases, an additional scallop for the celiac axis. Balloon-expandable bridging stent-grafts were implanted in all target vessels. The processed imaging data were totally anonymized; thus, no ethics committee approval was necessary.

The one-dimensional fluid-structure interaction dynamic model (1D) is based on the Navier–Stokes equations and the wall dilation equation to predict pulsatile flow, pressure, and arterial wall pulsatility. The 1D model consists of a system of cylindrical elastic tubes that mathematically represent blood flow through an extended portion of the arterial system. These tubes are interconnected, and the outflow from one segment serves as the inflow for the next one, with appropriate conditions applied at bifurcations [22,23]. The 1D equations can be derived directly from the Navier–Stokes equations, e.g., representation in polar coordinates, assuming axial symmetry. In the context of cardiovascular mechanics, 1D models have the potential to represent wave propagation through the vascular system, which is important in the aorta and larger systemic arteries. The model incorporates initial conditions by introducing pressure and flow waveforms in the ascending aorta. At the outlet of the 1D arterial tree, we apply more simplistic 0D-Windkessel models.

The 0D model incorporates a simplified three-element Windkessel fluid model to describe the peripheral vasculature, e.g., the smallest arteries and arterioles, and includes the peripheral resistance and the elasticity of the microvasculature to simulate and predict blood circulation. The model provides dynamic boundary conditions that predict the phase lag between flow and pressure and accounts for the effects of the propagation wave in the arterial system. Zero-dimensional models are systems of ordinary differential equations (ODEs) for the mathematical representation of microvascular regions.

The 1D mathematical model predicts the volumetric flow (q), pressure (p), and cross-sectional area (A) for each vessel of the arterial tree.

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial A}{\partial t}}+{\displaystyle \frac{\partial q}{\partial x}}=0,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial q}{\partial t}}+{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \frac{q^2}{2A}}\right)+{\displaystyle \frac{A}{\rho }}{\displaystyle \frac{\partial p}{\partial x}}=-{\displaystyle \frac{8\pi \mu q}{\rho A}},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & p(x,t)-p_0={\displaystyle \frac{4}{3}}{\displaystyle \frac{Eh}{r_0\left(x\right)}}\left(1-{\displaystyle \frac{A_0\left(x\right)}{\sqrt{A(x,t)}}}\right),\hfill \end{array}$$

(11)

where E is the Young’s modulus. At the inlet of the 3D aorta, a dynamic parabolic profile is enforced. Thus, spatially a parabolic profile that is changed at each time step based on a waveform obtained from the coupling of the 0D–1D mathematical model is applied. More specifically, the boundary conditions of the problem in this study are determined using the 0D–1D mathematical model, resulting in the inlet/outlet waveforms in each artery of the 3D reconstructed model [15], as depicted in Figure 1.

A localized three-dimensional (3D) dynamic mathematical model is introduced to study in detail the abdominal aorta, enabling the extraction of important hemodynamic parameters. Under these conditions, there are two fundamental variables, pressure p and velocity vector $\overline{q}$. Since the velocity vector has three directional components $(u,v,w)$, there are four degrees of freedom $(p,\overline{q})$, which together describe the physical state at each point of the flow. At the inlet of the 3D mathematical model, we applied parabolic waveforms obtained from the 0D–1D mathematical model, representing healthy characteristics. The pressure waveforms at the outlets of the 3D model were also derived from the mathematical model developed in 1D. The velocity profile at the outlets was described in the literature as almost parabolic during the systolic phase [13,15,24,25,26]. The no-slip condition was applied to the wall in all studied 3D cases.

The quantity time averaged wall shear stress (TAWSS) is defined as the average value of the magnitude of wall shear stress (WSS) vector along the cardiac cycle,

$$TAWSS\left(s\right)={\displaystyle \frac{1}{T}}{\int }_0^T\left|WSS(s,t)\right|dt,$$

(12)

where T is the duration of the cardiac cycle and s is the general surface coordinate.

Helicity, $H(s,t)$, is an index that quantifies the interplay between rotational and translational motion of blood, i.e., it has influence on the development of helical structures in the fluid domain. It is defined as follows:

$$H\left(t\right)={\int }_V\overline{v}(s,t)·\overline{w}(s,t)dV,$$

(13)

where $\overline{v}(s,t)$ is the velocity vector, $\overline{w}(s,t)$ is the vorticity vector, and V the fluid domain. Additionally, the rotational structures are visualized through the local normalized helicity (LNH) parameter [25,27,28].

$$LNH={\displaystyle \frac{\overline{v}(s,t)·\overline{w}(s,t)}{|\overline{v}(s,t)·\overline{w}(s,t)|}}=cos\phi ,$$

(14)

where $\varphi$ is the velocity and vorticity vectors angle.

LNH hemodynamic parameter as a function of space and time is an indicator of the helical structures’ intensity and their rotational direction [27,29,30]. When the absolute value of LNH is one, the flow is purely helical. Otherwise, when the value is zero, the flow is symmetric. The sign is of great importance as it dictates the right (+) or left-handed (−) direction of the helical structures. Therefore, the LNH values range between −1 and 1, as discussed in [15].

Model reconstruction and numerical solution. The computed tomography data obtained in this study included the pre- and postoperative scans of 3 patients for each technique (3 patients with FEVAR and 3 patients with BEVAR), with four stent-grafts and selected from a large cohort of patients. FEVAR and BEVAR patients, managed with 3 or 4 fenestrations/branches [celiac trunk (CT), superior mesenteric artery (SMA), left renal artery (LRA) and right renal artery (RRA)], that completed the initial month of follow-up (including CTA imaging), were considered eligible.

The 3D model of the fenestrated stent-graft system that includes the renal, superior mesenteric, celiac axis, and renal arteries was constructed using the computed tomography scan of the treated patient. Aiming to obtain the numerical solution, a procedure from the computed tomography scan to its transfer to the computing system must be followed. The image processing and reconstruction software, Mimics 15 (Materialise, Leuven, Belgium), was used for the reconstruction of the DICOM images into a 3D model. The aneurysmal sac was not taken into account in the reconstructed mathematical model [31,32]. An additional smoothing study was performed in Vascular Modeling Toolkit (VMTK), which led to an optimized smoothing factor of $0.05$. The computational grid of the reconstructed geometry was created with tetrahedral elements in the ICEM 15.1 (Ansys Inc., Canonsburg, PA, USA) software package.

The simulation has been set with $\nu =3.2\times {10}^{-6}$ m³ s⁻¹ the kinematic viscosity and $\rho$ = 1050 kg m⁻³, the blood density. The computational domain has a total of more than two million tetrahedral elements. To adequately study this problem, we performed three cardiac cycles. A small computational error of ${10}^{-4}$ was set as a stopping criterion, and we present the results of the final cardiac cycle avoiding any dynamic behavior of the first pulses [33]. A mesh sensitivity test was performed in areas of predicted disturbed flow; we obtained an optimal grid, choosing it regarding an error of less than $2\%$ in hemodynamic parameters.

In detail, we constructed three different grid configurations, the first composed of $1$ million elements, the second of $2$ million elements, and the third of $5$ million elements. Comparison with the second and third computational grids showed small differences between velocity and pressure in specific locations of interest, these differences were $2\%$. Additionally, a corresponding procedure was followed to correctly determine the time step. Simulations were performed for three cardiac cycles, the cardiac cycle was assumed to have a period of $T=1$ s, subdivided into 200 time steps, $\Delta t=0.005$, which gave robust numerical results compared to larger $\Delta t$. All 3D model simulations were performed using Fluent 15.1 (Ansys Inc., Canonsburg, PA, USA). The model ensures high-resolution analytics, including time averaged wall shear stress (TAWSS) and local normalized helicity (LNH).

3. Results

3.1. Two-Dimensional Bifurcation Results

The bifurcation geometry consists of a channel that leads to a division into two smaller and equal-diameter channels, also referred to as mother and daughter arteries. This simplification is aimed at understanding the flow mechanisms that arise from the branching network. In the last section of this study, a patient-specific analysis will be conducted, where conditions are not ideal and the geometry is not characterized as symmetric.

The computational domain is a rectangle with a length of $0.6$ and a height of $0.2$ dimensionless units. As depicted in Figure 2, the artery has a height of $H=0.1$ and length of $L=0.45$ dimensionless units. Additionally, they are presented with the boundary conditions that are applied in the solid boundaries in which the no-slip condition applies. A collocated grid has been constructed with rectangle cells. The angle of the daughter arteries is defined as $\varphi ={37}^{\circ }$. The rectangular grid has a size of $159\times 159$. A mesh sensitivity study has been conducted for both bifurcation/stenotic bifurcation. In this study, an error smaller than $2\%$ has been achieved for larger grids such as $259\times 259$. So, we chose the $159\times 159$ grid size to achieve optimal results in terms of accuracy and computational time.

The numerical solution obtained with the use of Matlab R22a (MathWorks Inc., Natick, MA, USA) has converged with the aid of the Levenberg–Marquardt algorithm, which suggests a powerful tool in the solving of the strongly non-linear system of equations. It is widely used in problems that utilize the coupling of the Finite Volume Method with Newton-like algorithms for pathological conditions such as aneurysmal geometries [34].

The boundary conditions for the problems under consideration are

$$\begin{array}{c}u\left(y\right)=\left\{\begin{array}{cc}0,\hfill & y\in [0.0,0.05)\cup (0.15,0.2]\hfill \\ u_{max}(-400y^2+80y-3),\hfill & y\in [0.05,0.15]\hfill \end{array}\right.,v=0,\mathrm{at}\mathrm{the}\mathrm{channel}\mathrm{inlet},\\ u=0,v=0,\mathrm{at}\mathrm{the}\mathrm{walls},\\ p=0,\mathrm{at}\mathrm{the}\mathrm{arteries}\mathrm{outlets}.\end{array}$$

(15)

These boundary conditions coupled with the system of Equations (2)–(4) provide the numerical solution, for u, v-velocity components, and pressure p.

Results in bifurcation. The bifurcation flow presents a velocity field of great interest. Both velocity and pressure fields are crucial for studying patient’s health and the development of pathological conditions. Blood flow in arteries is inherently pulsatile. However, due to the steady-state approach used in this study, a constant parabolic velocity profile is applied at the inlet. The Reynolds number $Re$ is 400, which roughly corresponds to the mean blood flow value between the systolic and diastolic phases.

In the channel section, the velocity profile remains parabolic, indicating a fully developed laminar flow similar to the Poiseuille flow. Right after the channel, the bifurcation divides into two daughter arteries of the same diameter. The flow near the bifurcation decreases and the two daughter branches receive symmetric flow. The flow in the daughter arteries near the stagnation point has a small thickness of the boundary layer compared to the boundary layer of the upper wall, as shown in Figure 3. The flow gradually becomes fully developed, and the velocity profile at the end of each branch is characterized as parabolic. As $Re$ increases, we observe smaller boundary layer thickness at the walls due to the increased momentum.

The contours of the u-velocity component do not present strong separation regions that could potentially lead to vortex creation. Higher angles between the daughter arteries could lead to recirculation zones. The narrowing in geometry suggests shear stress differences that could influence the arterial wall.

In Figure 4, the contours of the pressure field of the bifurcation are depicted. The pressure presents a gradual decrease similar to the fully developed flow in a channel, until the area of bifurcation. The highest pressure values are observed at the inlet of the channel. At the bifurcation, the pressure exhibits an increase due to the stagnation point. In the daughter branches, the pressure continues to decrease to reach a constant value at the outlet.

Another crucial parameter for biomedical flows is the wall shear stresses (WSS), given as the product of viscosity with the velocity gradient near the wall, ${\tau }_w=\mu {\left[{\displaystyle \frac{\partial u}{\partial y}}\right]}_w$. In Figure 5, the stresses on both walls are presented. The behavior is symmetric, as the bifurcation geometry is symmetric. The stresses remain constant in the channel section as the flow is fully developed. Right after this, the WSS increases for the bottom wall as it crosses the bifurcation and flows into the lower daughter artery. This suggests that fluid velocity near the bottom wall accelerates due to the narrowing of the channel, forcing the fluid to adjust, leading to a higher velocity gradient. The upper wall behaves identically to the bottom wall, with a different sign.

Results in Stenotic Bifurcation. In the case of the stenotic bifurcation, a stenosis is introduced close to the flow bifurcation to obtain vital changes that possibly affect the hemodynamic parameters. The geometric parameters are identical to those of the bifurcation in the previous subsection, with the addition of a $50\%$ stenosis of the artery, depicted in Figure 6. In this figure, the boundary conditions are described in Equation (15).

The stenosis impact is visible in Figure 7, which presents the u-velocity component. Stenosis drastically affects the flow behavior, as the highest velocity is observed in it. The maximum velocity value increases by almost $100\%$, and the flow reaches the value of 2. The overall behavior outside the stenosis is similar to that of a bifurcation. Upstream of the stenosis, the flow presents a typical fully developed laminar flow with a parabolic velocity profile. However, downstream of the stenosis the bifurcation is affected by the stenotic region, revealing an altered flow field from before.

Figure 8 presents the pressure field for $Re=400$. Initially, upstream of the stenosis, the pressure exhibits a decrease that almost resembles a typical channel flow until the region of stenosis. In the stenosis, the pressure shows a rapid decrease in contrast to the high velocity values in the stenotic region. Downstream of the stenotic section, the pressure fluctuates and presents an increase, which is due to the expansion of the geometry. This behavior drastically changes the pressure behavior compared to the previous case.

Finally, WSS, a critical factor in pathological conditions, is presented for both upper and lower walls in Figure 9. The WSS presents a strongly non-linear behavior due to the advanced geometric parameters of the geometry under consideration, such as a stenosis followed by a bifurcation. The stresses, at the bottom wall (blue color), at first present a steady behavior, and after this, rapidly increase due to the compression of the flow that leads to the increase in velocity and consequently to the increase in the WSS value. In the expansion section of the geometry, the WSS rapidly decreases. In this region, often recirculation zones occur that can appear via the change in sign in the WSS. In this case, vortices do not occur. However, a stenosis exceeding $50\%$ in addition to a larger Reynolds number could induce vortex formation.

Figure 10 presents the streamlines in the case of stenotic bifurcation. The colors below the streamlines represent the u-velocity component. These streamlines could be used to trace the trajectories of possible fluid particles, highlighting regions of flow acceleration and motion. This is a multiscale perspective, by connecting the Eulerian field with Lagrangian particle trajectories. The streamlines represent the collective motion of particles that respond to pressure and shear forces originating from upstream hemodynamics.

3.2. 3D Multiscale Patient-Specific Results

Simulations of the postoperative aorta after B/FEVAR grafts are presented, revealing the hemodynamics in these stent designs. In FEVAR, the flow remains undisturbed and can be characterized as smooth in the main body of the graft. In contrast, the visceral arteries exhibit more intense behavior. In the case of BEVAR, the velocity presents an increase in flow compared to FEVAR, as can be seen in Figure 11. Especially in the mesenteric artery, the flow shows a visible increase compared to BEVAR. The maximum values for FEVAR and BEVAR in systole are approximately 1.7 m/s and 2.5 m/s, respectively.

Rotational structures are of great interest in blood flow, and LNH is a reliable quantity to evaluate flow rotation. A threshold ($0.3$) has been established according to the literature [28,30,35]. The structures that represent helical schemes are distributed uniformly through the domains in both the BEVAR and FEVAR cases. The behavior of FEVAR LNH is depicted in Figure 12A, where the structures present a compact and smooth behavior, in contrast to BEVAR, Figure 12B. The maximum values and range of the two cases are identical, and finally, BEVAR case structures are more developed and have more rotational disposition (|LNH| $<0.3$). LNH of the visceral (secondary) arteries demonstrates a similar behavior.

Furthermore, the WSS as a significant parameter is presented in Figure 13A,B for both the FEVAR and BEVAR cases, respectively, at the peak systole and at the diastolic phase with a maximum value of $3\mathrm{Pa}$ for both designs. During the systolic phase in the BEVAR case, higher WSS values are observed compared to FEVAR, especially in the visceral arteries. At the diastolic phase, the WSS values are almost identical in both cases.

The TAWSS is shown in Figure 14 for both cases. In the BEVAR case, an overall increase is observed in the center section of the graft and in the visceral arteries. A large change is also observed in the RRA artery. The behavior of TAWSS in the FEVAR case ranges in intermediate values throughout the length. It shows relatively high values in SMA, RRA, and LRA at the proximal extent and proximal end. In BEVAR cases, the visceral arteries show increased values on most surfaces of the distal end. The postoperative results in both designs show several variations with a maximum value of 3 Pa.

Finally, a detailed table is introduced with all the hemodynamic parameters studied.

Table 1. Mean values for the hemodynamic parameters between FEVAR and BEVAR cohorts in the secondary arteries, SMA, RRA, and LRA, where MAP is the mean arterial pressure.

	FEVAR	BEVAR	FEVAR	BEVAR	FEVAR	BEVAR
	SMA	SMA	RRA	RRA	LRA	LRA
TAWSS-Mean (Pa)
Patient 1	2.0107	3.3955	1.8828	4.5400	1.7417	2.8088
Patient 2	2.4245	2.5005	2.7188	3.0423	2.8505	2.2069
Patient 3	2.6683	3.0862	2.3013	3.8802	2.0257	2.1344
Average	2.3668	2.9941	2.3019	3.9660	2.2059	2.3831
Flow Rate-Mean (mL/s)
Patient 1	1.20 $\times {10}^{-5}$	1.89 $\times {10}^{-5}$	7.77 $\times {10}^{-6}$	8.79 $\times {10}^{-6}$	4.91 $\times {10}^{-6}$	7.35 $\times {10}^{-6}$
Patient 2	4.53 $\times {10}^{-6}$	5.57 $\times {10}^{-6}$	5.77 $\times {10}^{-6}$	6.88 $\times {10}^{-6}$	4.76 $\times {10}^{-6}$	4.14 $\times {10}^{-6}$
Patient 3	6.20 $\times {10}^{-6}$	7.68 $\times {10}^{-6}$	5.15 $\times {10}^{-6}$	6.44 $\times {10}^{-6}$	3.48 $\times {10}^{-6}$	5.27 $\times {10}^{-6}$
Average	2.82 $\times {10}^{-6}$	1.08 $\times {10}^{-5}$	6.23 $\times {10}^{-6}$	7.44 $\times {10}^{-6}$	6.27 $\times {10}^{-6}$	6.27 $\times {10}^{-6}$
Pressure-MAP (Pa)
Patient 1	12,448.00	13,042.20	12,458.97	12,881.43	12,125.09	12,397.97
Patient 2	12,447.69	12,829.75	12,247.65	12,691.85	12,436.33	12,652.13
Patient 3	12,446.99	12,390.30	12,275.51	12,198.13	12,907.63	12,758.13
Average	12,447.89	12,754.75	12,327.38	12,590.47	12,489.68	12,569.74
WSS, peak systole (Pa)
Patient 1	15.7564	22.3136	15.1828	22.8823	13.0550	29.2930
Patient 2	20.5820	23.3455	17.4531	16.1189	19.5660	19.9550
Patient 3	23.3368	19.7350	18.6989	14.9496	25.6950	26.6720
Average	19.2211	23.3953	17.1111	18.9836	19.1054	24.0419

summarizes the mean values for the hemodynamic parameters, between the FEVAR and BEVAR cohorts in the secondary arteries (SMA, RRA, LRA).

4. Discussion

The present work provides an analysis of blood flow in bifurcated stenotic and non-stenotic arteries for $Re=400$. For the channel upstream of the bifurcation region, the velocity and pressure fields exhibit a typical Poiseuille flow that is disturbed by the appearance of the bifurcation. A notable difference is observed near the stagnation point of the bifurcation, due to the collision of fluid particles with the arterial walls that drastically increases the WSS values. The flow does not present recirculation regions that could potentially lead to thrombus formation.

The addition of stenosis increased the velocity magnitude and led to a notable decrease in pressure in the stenotic region. The WSS showed an increase of more than $100\%$ in the stenosis region that could have an undesirable effect on the arterial walls. Furthermore, the flow quantities $u,p$ and $WSS$ near the bifurcation were altered by adding the stenosis.

FEVAR and BEVAR are considered optimal treatment options for complex aortic aneurysms. In this study, we compared the postoperative hemodynamic effects on the main and secondary vessels (celiac trunk–CT, superior mesenteric artery–SMA, right renal artery–RRA, left renal artery–LRA) after endovascular aneurysm repair with FEVAR and BEVAR. The patient-specific computational approach offered valuable insights. These findings are of clinical significance and could play a role in selection of the right treatment, minimizing complications in the future [14,33]. A thorough comparison of postoperative hemodynamic characteristics between FEVAR and BEVAR was performed, highlighting the importance of visceral arteries. Both techniques are considered reliable, resulting in similar outcomes, which are close to normal (healthy) levels [14,17,33].

The postoperative results show changes in the main lumen and in the visceral arteries in fluid flow [14]. The behavior of the key hemodynamic parameters is different in each of the two cases, providing a significant flow improvement [13,14,16,17].

Flow, pressure and WSS: Postoperatively, the hemodynamic state shows changes in flow within the main lumen and in the visceral arteries. Each of the two techniques has a different response to hemodynamic indicators, providing significant flow improvement. Postoperative variations in flow and pressure show a correlated common course. In BEVAR, pressure in CT, SMA, RRA, and LRA has an increasing trend, with subtle differences, compared to FEVAR. Relatively greater incremental difference is seen in the CT artery compared to FEVAR. Flow also increases in CT, SMA, RRA, and LRA, but decreases in the CT artery compared to FEVAR. Postoperatively, due to the reconstruction of the geometry, there was a significant reduction in pressure overall. Specifically, in the case of BEVAR the pressure fluctuates at higher levels compared to FEVAR [14]. For BEVAR at systolic peak, the flow rate was higher than in FEVAR; between the renals it ranged moderately, but the most intense variation occurred in the mesenteric artery. Postoperatively, flow in CT, SMA, RRA, and LRA with both techniques remodeled to approach normal (healthy) hemodynamic characteristics [33]. In BEVAR cases, the WSS is high overall, in the whole structure and in the secondary arteries. In the FEVAR case, WSS is also high in the visceral arteries, but shows slightly lower values in the main graft body. Almost identical characteristics have been observed in previous simulations conducted in FEVAR cases [13,15].

LNH and TAWSS: Large LNH values are associated with regular flow circulation, inhibiting blood cell adhesion to the arterial wall and reducing thrombus formation [28]. LNH in FEVAR cases presents uniform and compact structures throughout the cardiac cycle, which is also validated in previous studies [13,15]. In the BEVAR cases, in the diastolic phase, there is more pronounced redistribution of helical structures. Overall, values of TAWSS are higher than in the BEVAR case and more intense in the RRA artery; this is cross checked with the work in [14]. They concluded that the geometric morphology of the renal arteries in the patient was the reason for the variations observed between the FEVAR and BEVAR cases. Thrombus deposition and atherogenesis are associated with abnormally high or low TAWSS levels [14,17].

Flow rate, geometric characteristics and flow reversal: In the BEVAR cases, the flow rate at the systolic peak was slightly higher than in the FEVAR. More specifically, between renals it ranged approximately the same, but the most significant variation is observed in the mesenteric SMA artery. In BEVAR, there is a section between CT, renal arteries, and iliacs where the flow is relatively high. Postoperatively, flow on CT, SMA, RRA, and LRA with both techniques remodeled to approach normal (healthy) hemodynamic characteristics. The fundamental difference between BEVAR and FEVAR with the larger cross-sectional area of fenestrations may explain the larger values in the velocity field. In FEVAR procedures, the fenestrations are vertically aligned with the custom-made openings, while in BEVAR procedures, they are incorporated into built-up side branches within the main graft segment, consisting of devices that combine visceral side branches with fenestrations [14]. Significant changes in the geometry of these side branches (whether FEVAR or BEVAR) could negatively impact blood flow behavior. In the BEVAR technique, longer bridging stents facilitate blood flow expansion before it reaches the visceral artery [17]. This essential difference might also contribute to the creation of recirculation regions in the FEVAR case at the entrance of the bridging stents. In the BEVAR case, the recirculation regions are more intense in the CT and LRA arteries, leading to an increase in the reversed flow rate. This may be promoted by the fact that there is a flow decrease in this artery.

Hemodynamic variations provide first-hand information for alterations in the postoperative setting, and this study showed that despite differences in both techniques, they seemed to provide close to similar outcomes. In addition, the patient-specific computational approach enables better information and tools for revised hemodynamics after surgery and could affect future decision-making in patients who may be candidates for either technique.

The patient-specific computational approach provided useful information. These findings are clinically important and may in the future be a factor in the selection of appropriate treatment and the reduction of complications. However, remodeling of the visceral arteries after stenting might cause local hemodynamic disturbances with possible clinical consequences. Computational approaches have the potential to analyze local high resolution hemodynamic conditions.

Overall, the two EVAR techniques (FEVAR, BEVAR) increase the life expectancy of patients with complex aortic aneurysms. Based on the CFD results, clinicians can evaluate the applications of the two methods, follow the appropriate treatment strategy for each patient (patient-based approach) in the treatment of complex aneurysms, and predict postoperative complications.

5. Conclusions

In this study, the geometric effects on blood flow are analyzed in both bifurcated and stenotic bifurcated cases. The coupling of the Finite Volume Method with the Levenberg–Marquardt algorithm ensures convergent and robust results. Geometry plays a crucial role in flow behavior, with particular interest in the stresses observed in both cases. Specifically, the geometry of the stenotic bifurcation exhibits higher values of wall shear stresses compared to the standard bifurcation, highlighting the danger to patient health from such a pathological condition. In addition, significant pressure gradients observed in the stenotic case may contribute to vascular complications.

Furthermore, a multiscale mathematical model is employed, coupling the 0D Windkessel with 1D boundary conditions waveforms and finally the 3D simulations to provide valuable insights into postoperative aortic blood flow. Some important and influential quantities—such as velocity, LNH, WSS, and TAWSS—are examined in patient-specific aortas in cases of advanced stent devices BEVAR and FEVAR. These state-of-the-art grafts are considered the best for the treatment of abdominal aortic aneurysms (AAAs) and are worth analyzing by CFD. The two grafts provide similar results in most parameters and improve toward normal (healthy) hemodynamics. The FEVAR exhibits smaller values of velocity and more uniform, smooth helicity compared to the BEVAR. The visceral arteries modeled through the multiscale model present higher values in BEVAR compared to the FEVAR cases. The 3D patient-specific aortas used enhance the clinical relevance of this study, providing valuable guidance for personalized treatment strategies. This could lead in the future to technological refinement of more specific branch endografts to be used in F/BEVAR.