Comparison of Rigid-Wall Computational Fluid Dynamics and Flexible-Wall Fluid-Structure Interaction in Descending Thoracic Aorta Aneurysm

Abstract

Currently, Computational Solid Mechanics (CSM) and Computational Fluid Dynamics (CFD) simulations are not enough to correctly estimate the different physical characteristics found in the human cardiovascular system. As an alternative to individual simulations, Fluid Structure Interaction (FSI) simulations can yield more accurate physical quantities. In this study a comparison between rigid-wall CFD of a thoracic aorta affected by an aneurysm and the FSI of the Descending Thoracic Aortic Aneurysm (DTAA) itself was performed. The 18-year-old patient-specific geometry of the aorta and its branches was based on the National Institutes of Health public database. A patient-specific pulsatile blood flow waveform and a pressure three-element Windkessel model were set for boundary conditions. Parameters such as wall pressure, velocity distribution, Wall Shear Stress (WSS), Time-averaged Wall Shear Stress (TAWSS), Oscillatory Shear Index (OSI), wall displacement and Von Mises Stress (VMS) were investigated. The research shown that blood flow in the aorta is strongly affected by the onset of the aneurysm, which causes recirculation and uneven flow within the aneurysmal bulge. The results highlight that rigid-wall CFD, which cannot capture wall deformation and aneurysm compliance, leads to an overestimation of velocity, WSS, and TAWSS by 15, 21, and $32\%$ respectively, compared to FSI during the systolic peak; furthermore, a key novelty is represented by the slight underestimation of pressure during the systolic peak, an aspect not previously detailed in the DTAA literature.

1. Introduction

The cardiovascular system exhibits highly complex fluid–structure interactions, where blood flow, pressure distributions, and wall shear stresses play central roles in both physiological function and the disease development [1,2,3]. Because many cardiovascular disorders (such as pulmonary hypertension, aneurysms, coronary artery diseases, etc.) are strongly influenced by local hemodynamic conditions, understanding the flow characteristics is essential for improving diagnosis, risk assessment, and treatment planning [4]. However, direct in vivo measurement of detailed flow fields remains challenging due to limitations in spatial resolution, invasive techniques, and inter-patient variability [5,6,7,8].

Computational Fluid Dynamics (CFD) has therefore emerged as a powerful tool for investigating cardiovascular hemodynamics with high fidelity [9]. By solving the governing equations of fluid motion within patient-specific domains, CFD enables researchers to explore local pressure gradients, velocity fields, and wall shear stress distributions that are otherwise difficult or impossible to measure experimentally [10,11,12]. Recent studies have demonstrated the value of CFD in analyzing pathological conditions, helping to quantify abnormal pressure loads and flow disturbances [9,13,14,15,16].

The integration of CFD with medical imaging and patient-specific modeling is transforming cardiovascular research. It allows clinicians and engineers to simulate disease scenarios, evaluate surgical interventions, and design personalized treatment strategies. As computational methods and imaging technologies continue to advance, CFD is becoming an indispensable component of modern cardiovascular biomechanics, offering a non-invasive, cost-effective, and highly detailed window into the dynamics of blood flow.

However, an accurate description of cardiovascular hemodynamics often requires not only modeling blood flow but also accounting for the mechanical response of vascular tissues, making Fluid-Structure Interaction (FSI) a fundamental component of advanced numerical simulations [17,18,19,20]. Blood vessels continuously deform under pulsatile pressure loads, and this deformation in turn alters the flow field, creating a bidirectional coupling that rigid-wall assumptions do not capture [11]. By integrating CFD with structural mechanics, FSI models offer a more realistic and physiologically accurate representation of the cardiovascular system, potentially enabling improved predictions of disease progression and offering more reliable support for medical evaluations.

Several numerical strategies have been developed to tackle the challenges FSI in hemodynamics [4]. The Arbitrary Lagrangian-Eulerian (ALE) framework [21,22] has become the most widely adopted approach due to its ability to accurately track the fluid-solid interface and its computational efficiency in partitioned solvers. However, it is well-known that ALE is primarily suited for small to moderate structural deformations (less than $10\%$) [23], as excessive displacements can lead to severe mesh distortion and to numerical instabilities.

In this context, more advanced numerical frameworks have recently emerged to overcome these limitations, especially for strongly coupled problems or extreme deformations. These include monolithic fully implicit formulations based on Eulerian representations of the elastic structure [24]. These approaches are particularly effective in handling strong coupling effects and avoiding the ’added mass’ numerical instabilities typical of partitioned schemes when dealing with lightweight structures in incompressible flows [25]. Despite these advancements, for the physiological range of the thoracic aortic wall displacements, which is the focus of the present study, the ALE framework remains a reliable and accurate choice [4,26,27].

In addition, there is also growing interest in the integration of machine learning (ML) and data assimilation techniques within the field of computational biomechanics [28]; however, high-fidelity CFD and FSI simulations remain indispensable tools, particularly for the study of descending thoracic aortic aneurysms (DTAA). To date, in fact, the literature concerning the application of ML models specific to DTAA is still limited compared to other vascular regions [29,30,31]. Direct numerical simulation is essential for generating the reference datasets required for the training and validation of future data-driven models [32,33], ensuring that they are physically consistent.

In various studies, FSI techniques have been adopted to investigate blood flow in different cardiovascular vessels, providing estimates of the main fluid-dynamic fields as well as the distribution of mechanical stresses within the vascular tissues [26,34,35,36]. Multiple studies demonstrate the clinical relevance of FSI simulations of aortic blood flow and aortic aneurysms, providing relevant parameters such as wall stresses, local wall deformations, pressure, velocities, while also enabling the visualization of the flow pattern in the aneurysm zone [37,38,39,40,41,42]. In addition, FSI and finite element-based computational techniques are essential for the analysis of native and pathological valve haemodynamics, as well as for the design and optimization of implantable heart valves [43,44]. Recent studies have investigated the fluid–structure interaction of healthy aortic valves and the surrounding flow field [45], the influence of bicuspid aortic valve cusp fusion patterns on aortic wall shear stress distributions [46], and patient-specific computational haemodynamics for pulmonary valve replacement [47]. These works demonstrate the relevance of numerical modeling for quantifying wall shear stress and related haemodynamic indicators in valvular flows. With respect to these contributions, the present study focuses on a comparison between classical CFD and FSI simulations for blood-flow in cardiovascular system.

In the last year, only a few studies have focused on comparing fully coupled fluid–structure interaction (FSI) of Descending Thoracic Aortic Aneurysm (DTAA) models with rigid-wall CFD and other simplified approaches for patient-specific aortic simulations [37,48,49,50,51,52]. These investigations consistently demonstrate that while rigid models can approximate global metrics, such as pressure and flow waveforms, they fail to capture compliance-driven local phenomena [53], including wall shear stress and secondary flow patterns, which are critical for accurate risk assessment [54,55].

In this context, this work aims to contribute to this topic by comparing rigid-wall CFD and flexible-wall FSI simulations of blood flow in a patient-specific descending thoracic aortic aneurysm, highlighting the capability of FSI simulation to capture wall compliance and the role of the aneurysmal sac as a flow accumulation zone. We selected a case study of a patient-specific aortic geometry, with pulsatile blood flow waveforms. For the pressure outlet boundary conditions, we rely on a three-element Windkessel model [56].

2. Materials and Methods

2.1. Geometry

For this study, a publicly available three-dimensional model of the thoracic aorta was obtained from the National Institutes of Health (NIH) database [57], ensuring reproducibility and accessibility. The model, shown in the Figure 1, represents an 18-year-old patient with a descending thoracic aortic aneurysm. The domain for the FSI simulation is the downstream part of that in the CFD simulation, more precisely from “Cut for FSI” to Outlet in Figure 2a.

2.2. CFD Set-Up

To ensure consistency between the rigid-wall CFD and flexible-wall FSI analyses, an initial CFD simulation was conducted on the full aortic geometry. The resulting flow data were mapped onto a newly defined inlet section (Cut for FSI), which was used as the inlet boundary condition for the FSI simulation, as depicted in Figure 2a.

The mesh was built with cfMesh [58] and consisted of 861201 cells, and it is shown in Figure 2b,c. The adopted computational mesh was determined through a systematic grid sensitivity analysis, described in detail in the Section 2.2.3, employing the Grid Convergence Index (GCI) method, ensuring mesh-independent numerical results [59].

2.2.1. CFD Numerical Simulation

The fluid problem was modeled using the Navier-Stokes equations, considering an incompressible, laminar fluid. Blood, as is well known, is a non-Newtonian and shear-thinning fluid [60] due to its complex composition and the interactions between its components. In particular, when the shear rate exceeds 100 s⁻¹ [1,13] and in the case of large vessel diameters relative to red blood cell size, it is commonly accepted in the literature to consider blood as a Newtonian fluid [4,14,61,62]. It should be noted, however, that in areas of low shear rate, such as the aneurysmal sac Newtonian models can lead to variations in haemodynamic parameters such as WSS [41,63,64]. In this particular study, the focus is on the aorta where the diameters are larger than 0.1 mm, allowing blood to be treated as a Newtonian fluid

$$\begin{array}{cc}\hfill {\rho }_f\left[{\displaystyle \frac{\partial {\mathbf{u}}_f}{\partial t}}+({\mathbf{u}}_f·\nabla ){\mathbf{u}}_f\right]+\nabla p-\mu \Delta {\mathbf{u}}_f& =0\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \nabla ·{\mathbf{u}}_f& =0\hfill \end{array}$$

(2)

where ${\mathbf{u}}_f$ is the fluid velocity field, p is the pressure and ${\mu }_f$ is the dynamic viscosity.

Considering the sections with the largest diameter and peak velocity, $S_1$ and $S_3$, depicted in Figure 3, the Reynolds number ($Re$) calculated during the systolic peak is found to be 1833 and 4072 respectively. Although the value of the $S_3$ section is in the transition regime, the laminar flow model is still valid. Indeed, considering the critical Reynolds model ($R{e}_c$) in the thoracic aorta [65,66]

$$R{e}_c=k\alpha$$

(3)

where k is a constant between 250 and 1000 and $\alpha$ is the Womersley number which is equal to $21.9$ for section $S_1$ and 10 for section $S_3$. The resulting range of values, respectively for $S_1$ and $S_3$, is 5475 to 21,900 and 2500 to 10,000, which is higher than the calculated Re numbers. Furthermore, the calculated Re values are the maximum values found in the critical section of our domain and occur only during small part of the cardiac cycle.

The blood was modeled with a density of 1060 kg/m³ and a kinematic viscosity [9,67] of $4\times {10}^{-6}$ m²/s. Implicit first-order Euler method was used for temporal discretisation to ensure high numerical stability, which is critical in FSI problems with a high ratio of added mass to structural mass, such as this one. A variable time step of the order of $1\times {10}^{-3}$ s was chosen. In terms of spatial discretisation, second-order schemes were chosen; specifically, the Gauss-linear scheme for the gradient, the Gauss-linear-Upwind scheme for the divergence, and the Gauss-linear-corrected scheme for the Laplacian term. The PIMPLE algorithm was adopted to compute the solution, with the end of the cycle controlled by tolerances on velocity and pressure residuals of $1\times {10}^{-8}$ and $1\times {10}^{-7}$, respectively.

2.2.2. Boundary Conditions

A three-element Windkessel model was used to model the pressure boundary condition for the RSA, RCCA, LCCA, and LSA outlets. They represent the Right Subclavian Artery, the Right Common Carotid Artery, the Left Common Carotid Artery, and the Left Subclavian Artery. The following equation describes the Windkessel model:

$${\displaystyle \frac{dP\left(t\right)}{dt}}={\displaystyle \frac{(Z+R)}{CR}}Q\left(t\right)+Z{\displaystyle \frac{dQ\left(t\right)}{dt}}-{\displaystyle \frac{P\left(t\right)}{CR}}$$

(4)

where Q is the flux and the R, Z and C parameters are specific to each patient and are related to mean aortic pressure and flow rate [56,68]; their values are reported in

Table 1. Resistance and compliance values used.

	R $\left[\frac{\mathbf{Pa} \times \mathbf{s}}{{\mathbf{m}}^3}\right]$	Z $\left[\frac{\mathbf{Pa} \times \mathbf{s}}{{\mathbf{m}}^3}\right]$	C $\left[\frac{{\mathbf{m}}^3}{\mathbf{Pa}}\right]$
LCCA	$2.18\times {10}^8$	$6.58\times {10}^8$	$2.42\times {10}^{-9}$
LSA	$1.74\times {10}^8$	$5.24\times {10}^8$	$3.04\times {10}^{-9}$
RCCA	$1.71\times {10}^8$	$5.14\times {10}^8$	$3.09\times {10}^{-9}$
RSA	$1.32\times {10}^8$	$3.98\times {10}^8$	$4.00\times {10}^{-9}$

and can be defined from patient-specific haemodynamic measurements, mainly cardiac output and mean aortic pressure, as reported in the literature [56,68,69,70]. In particular, the total vascular resistance is distributed between $R_p$ and $R_d$, with $R_p$ set to 10% and $R_d$ to 90% of the overall resistance, following previous studies [13,69]. The three-component Windkessel model provides an accurate description of the role of aneurysmal compliance and vascular resistance. In particular, the capacitive element accurately simulates arterial dilation during systole, where energy is stored and subsequently released during diastole when the heart relaxes, thereby maintaining proper blood flow and pressure. The resistive elements, on the other hand, allow for the simulation of the resistance encountered by the blood flow as it moves in small and large vessel. The above boundary condition was implemented as a Dirichlet condition for pressure in the OpenFOAM software, version 2012 [71]. The ordinary differential equation (ODE), expressed by Equation (4), was solved using backward discretization method.

A pulsatile pressure waveform [72] was prescribed as a Dirichlet boundary condition at the descending aortic outlet, referred to as Outlet, as illustrated in Figure 4a. The volumetric flow rate, Figure 4b, was imposed at the Inlet as the boundary condition.

2.2.3. Mesh Sensitivity Analysis

Three grids, M1, M2 and M3, ranging from the finest to the coarsest (

Table 2. Mesh characteristics, where $h_{average}$ refers to average size of edge length.

Mesh	n° Cells	WSS $\left[\mathbf{Pa}\right]$	Flow Rate $\left[\frac{\mathbf{mL}}{\mathbf{s}}\right]$	${\mathbf{h}}_{\mathbf{average}}\left[\mathbf{mm}\right]$
M1	2,949,145	$6.861$	$208.806$	$0.359$
M2	861,201	6.844	$208.880$	$0.542$
M3	289,731	$6.739$	$208.909$	$0.779$

) were analyzed to assess mesh sensitivity. The GCI was evaluated for the WSS in the region highlighted by the box in Figure 3, corresponding to the DTA section downstream of the aneurysm at peak systole. In addition, the GCI for the volumetric flow rate was computed at section $S_3$ in Figure 3. To ensure an appropriate calculation of WSS, the flow near the wall needs to be properly captured through a y+ value that should be less than 2 [41,42,73]. In our case, a posterior check shows that for mesh $M_2$, which has a first layer thickness of $0.05$ mm, the y+ value is less than 2 during the systolic peak, which is the most critical point for this parameter due to the highest velocities. The section chosen is part of the FSI simulation domain where, due to wall deformation (which we will discuss later), lower velocities are observed; this results in lower $y^{+}$ values, which ensure an even more accurate WSS calculation, given the consistency of the fluid mesh used in both models. For the reasons explained earlier and due to high computational cost of FSI simulation, it was decided to carry out the sensitivity analysis exclusively for the CFD simulation.

The results of the grid sensitivity analysis are shown in

Table 3. Grid Convergence Index and the order of convergence for the WSS and Flow rate.

Mesh	${\mathbf{GCI}}_{\mathbf{wss}}$	${\mathbf{GCI}}_{\mathbf{Flow}\mathbf{rate}}$	Refinement Factor $\mathbf{r}=\frac{{\mathbf{h}}_{\mathbf{coarse}}}{{\mathbf{h}}_{\mathbf{fine}}}$
M1–M2	$0.040\%$	$0.021\%$	1.51
M2–M3	$0.331\%$	$0.010\%$	1.44

. As shown, the GCI values associated with the medium mesh are below $0.04\%$ for WSS and $0.02\%$ for flow rate when compared with the finest mesh, indicating negligible discretization errors. Consequently, the medium mesh (M2) was deemed sufficiently accurate and was adopted for all subsequent simulations.

2.3. FSI Set-Up

The FSI simulation was performed only on the portion of the thoracic aorta affected by the aneurysm. For the fluid domain, the geometry and mesh shown in the Figure 5 were used.

The framework used consists of OpenFOAM [71], CoCoNuT [74], and Abaqus [75]. The first solver was used for the fluid-dynamic problem, while the last was used for the structural one; CoCoNuT, serves as the coupling interface between the fluid and structural solvers.

The CFD simulation results were used to extract the velocity values at the Cut for FSI cross section, which were then mapped and used as inlet boundary condition for the FSI simulation. In the FSI case, the fluid domain mesh consists of 447,623 cells and is depicted in Figure 5b,c.

The mesh for the aortic walls, shown in Figure 6, was generated by extruding the surface defining the fluid domain in the normal direction. The wall thickness is uniform and equal to 2 mm [11,76], discretized into 3 layers of hexahedral cells. This number of layers was chosen to ensure accuracy and acceptable computational time for the FSI simulation, as described in [26]. The solid domain mesh consists of 34,632 hexahedral cells and was created using ICEM software, version 2024 R1.

2.3.1. FSI Numerical Simulation

The FSI calculation consists of a fluid domain ${\mathsf{\Omega }}_f$ and a solid domain ${\mathsf{\Omega }}_s$. The fluid domain was modeled using the same equations and method described in the Section 2.2.1. The aortic wall consists of the intima, media and adventitia, separated by an elastic membrane [77]. The intima is the thinnest inner layer of the three, the media is the middle layer, and the adventitia is the outer, elastic layer. Such a complex tissue would require, as a constitutive model, the use of anisotropic models such as the Holzapfel-Gasser-Ogden (HGO) model [78,79]. However, most of the literature uses linear and non-linear isotropic models [4,37,41,76], despite recognising their limitations in terms, for example, of stress and strain evaluation [80]. In this study, the solid domain was modelled using a Neo-Hookean hyperelastic model [81], cf. Equation (5), with a Poisson’s ratio of $0.45$, a value close to incompressibility, a Young’s modulus of $1$ MPa and a density of $1000{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}$. The material model is based on the following equation

$$\varphi (\mathit{B},J)={\displaystyle \frac{\mu }{2}}\left[J^{-{\displaystyle \frac{2}{3}}}tr\left(\mathit{B}\right)-3\right]+{\displaystyle \frac{\kappa }{2}}\left[{\displaystyle \frac{1}{2}}(J^2-1)-lnJ\right]$$

(5)

where $\varphi$ is the strain energy density function, $\mathit{B}$ is the Cauchy-Green deformation tensor, J is the Jacobian determinant of the deformation gradient, $\mu$ is the second Lamé parameter and $\kappa$ is the bulk modulus. As mentioned earlier, in the absence of patient-specific clinical data non-linear isotropic material, even though the effects of anisotropy and the multilayered nature of the aortic wall are neglected, provides a reasonable compromise between model complexity and computational cost, allowing the investigation of main haemodynamic parameters and wall deformations [4,26,42,82].

The motion of the solid domain is governed by the balance of linear momentum:

$${\rho }_s{\displaystyle \frac{{\partial }^2{\mathbf{d}}_s}{\partial t^2}}=\nabla ·{\mathit{\sigma }}_s+{\rho }_s{\mathbf{b}}_s\mathrm{in}{\mathsf{\Omega }}_s,$$

(6)

where ${\rho }_s$ is the solid density, ${\mathbf{d}}_s$ is the displacement field, ${\mathit{\sigma }}_s$ is the Cauchy stress tensor of the solid, and ${\mathbf{b}}_s$ is the body force per unit mass. The solid stress tensor ${\mathit{\sigma }}_s$ is obtained from the Neo-Hookean strain-energy density function reported in Equation (5).

FSI simulations require that two equilibrium conditions [36] be satisfied at the fluid-structure interface ${\Gamma }_{\mathrm{FSI}}$. The first is the kinematic condition:

$${\mathbf{u}}_f={\displaystyle \frac{d{\mathbf{d}}_s}{dt}}\mathrm{on}{\Gamma }_{\mathrm{FSI}}$$

(7)

where ${\mathbf{d}}_{\mathbf{s}}$ is the displacement vector of the solid domain at fluid–structure interface, and ${\mathbf{u}}_{\mathbf{f}}$ is the fluid velocity evaluated on the fluid side of the interface.

The second condition is the dynamic equilibrium at the interface:

$${\mathit{\sigma }}_f·{\mathbf{n}}_f=-{\mathit{\sigma }}_s·{\mathbf{n}}_s\mathrm{on}{\Gamma }_{\mathrm{FSI}}$$

(8)

where ${\mathit{\sigma }}_f$ and ${\mathit{\sigma }}_s$ are the fluid and solid stress tensors, respectively, while ${\mathbf{n}}_f$ and ${\mathbf{n}}_s$ are the unit normal vectors pointing outward from the corresponding subdomains.

The adopted FSI strategy is partitioned strongly coupled, and the implicit Interface Quasi-Newton with Inverse Jacobian (IQNI) algorithm was employed to perform the coupled calculations [36]. An absolute convergence criterion based on the $L_2$ norm (of order 2) of the displacement residual at the interface has been imposed. The maximum number of FSI coupling iterations was set to 20 and the convergence is achieved when the residual value falls below the set tolerance, equal to $1\times {10}^{-5}$ m. A fixed time step of $1\times {10}^{-3}$ s was adopted for the FSI simulation, providing a reasonable computational cost whilst ensuring numerical stability, particularly given the very similar densities of the fluid and solid, incompressible fluid and the nearly incompressible behavior of the solid material [19]. The simulation was carried out up to the third cycle, which was taken as a reference for the physiological parameters analysed [37,42,48,49,83]. By the third cycle, the simulation reach stability (cf. Figure 7), as the difference between the pressures, calculated using the Equation (9), at the start of the cardiac cycle is less than $2\%$.

$$difference={\displaystyle \frac{P_{n+1}-P_n}{P_n}}<2\%$$

(9)

where $P_n$ denotes the initial average pressure at the nth cycle at section $S_3$.

The FSI simulation was performed using 22 cores, and it required about 500 core-hours to complete one cardiac cycle.

Using the same number of cores, the CFD simulation required about 100 core-hours. The FSI simulation required a computational cost 5 times greater than the CFD.

2.3.2. Boundary Condition

For the solid domain, the Outlet was constrained with zero displacement, whereas a radial displacement boundary condition was prescribed at the Inlet to permit radial deformation consistent with physiological wall compliance. Radial deformation, defined with respect to a local reference system, is not natively supported in Abaqus for non-axisymmetric geometries. Therefore, a custom Python, version 3.10, script was developed to compute the appropriate local reference system and to correctly apply this boundary condition.

Regarding the fluid domain, the pressure boundary condition shown in Figure 4a was imposed at the Outlet, while a velocity boundary condition was prescribed at the Inlet. The inlet velocity field, mapped from the rigid-wall CFD simulation, was appropriately scaled to account for the time-varying inlet area (Figure 2a) to maintain the exact same flow rate curve. This velocity vector rescaling was necessary due to the significant radial deformation of the inlet section.

3. Results and Discussion

Rigid-wall CFD and FSI simulations of a thoracic aorta affected by an aneurysm were compared to understand the numerical differences between the various physical quantities of interest, such as velocity, pressure, WSS, TAWSS, OSI and VMS. The following section presents these quantities in an orderly manner, highlighting the differences between CFD and FSI.

3.1. Velocity

In Figure 8, some differences can be noted between the two types of simulations. At section $S_3$ (cf. Figure 3 for the location), the local maximum velocity is reached at the systolic peak, considered to be reached at the maximum flow peak with reference to the Figure 4b, $1.204$ m/s in the CFD simulation and $1.045$ m/s in the FSI simulation, corresponding to a increase of about ≃15% relative to the FSI result. During the diastolic phase, a maximum local velocity of $0.472$ m/s for CFD and $0.365$ m/s for FSI is obtained, with an increase of about ≃29% relative to FSI simulation. The maximum velocities are both reached in the DTA section at systolic peak. The result is consistent with the physics of the problem, as the cross-section area of the DTA part, in the FSI case, increases due to the internal pressure (Figure 3). In the FSI simulation, the area of the slice $S_3$ is equal to $2.34$ cm², while in CFD it is equal to $2.04$ cm², an increase of ≃12%. If we consider the average velocity calculated at systolic peak downstream of the aneurysm, represented by the box in Figure 3, as expected, the velocity is greater in the CFD case than in the FSI case, specifically $1.043$ m/s vs. $0.885$ m/s. During the diastolic phase, the velocity reaches $0.298$ m/s for the CFD simulation and $0.216$ m/s for the FSI simulation. These velocity values are consistent with the current literature in the case of DTAA [11,41].

In both cases, given the presence of the aneurysm, there is strong recirculation of the blood within the bulge, as can be seen in Figure 9. The flow becomes disorganized and recirculation causes blood particles to remain in prolonged contact with the surface of the aneurysm, leading to potential platelet deposition on the surface [84]. Platelet deposition on the surface of the lumen contributes to thrombus formation.

3.2. Pressure

With regard to pressure, particularly with reference to Figure 10, a slightly higher pressure is observed in the FSI calculation compared to the CFD simulation during the systolic peak. This behaviour is due to the compliance of the deformable walls. In fact, as we can see from Figure 11, while in the CFD case the flows calculated in sections $S_1$ and $S_2$ coincide with the Inlet flow, in the FSI case there is a delay in the flow peak due to the deformation of the walls, which initially stores part of the inflowing volume. The aneurysm therefore acts as a flow accumulation zone until it reaches its maximum expansion limit. Thereafter, the pressure increases due to the incoming flow pushing the fluid already present inside the aneurysmal sac, against the distal narrowing, and due to the reaction of the deformable wall on the fluid, which releases the deformation energy previously stored. The characteristics of the material, in particular its low stiffness and uniform thickness, tends to accentuate the phenomenon just described. The deformation of sections, $S_1$ and $S_2$, shown in Figure 3, vary differently between the beginning of the cardiac cycle and the systolic peak by, $8\%$ and $5\%$ respectively, further confirming the accumulation zone behaviour exhibited by the aneurysm. Considering the diastolic phase, it can be seen that the percentage of deformation between sections $S_1$ and $S_2$ are substantially equal, thus causing a decrease in overall pressure in the FSI simulation compared to the CFD simulation. The average pressure across the entire domain at systolic peak is 14,797 Pa for the CFD calculation and 14,899 Pa for the FSI simulation. During the diastolic phase, instead, the CFD and FSI simulations yield values of 10,594 Pa and 10,464 Pa respectively. In both cases, low-pressure areas can be observed in the proximal and distal necks of the aneurysm, in agreement with [11,41].

3.3. Wall Shear Stress and Time Averaged Wall Shear Stress

WSS is one of the most important and analyzed parameters in this type of simulation, both in CFD and FSI [4]. WSS is defined as the tangential component of traction vector acting on the wall [37,42] and is calculated using Equation (10)

$$\mathbf{wss}=\mathbf{\tau }-(\mathbf{\tau }·\mathbf{n})\mathbf{n}$$

(10)

where $\mathbf{\tau }=\mu (\nabla \mathbf{u}+\nabla {\mathbf{u}}^T)\mathbf{n}$ is the traction vector due to shear stress and $\mu$ is the dynamic viscosity. In particular, a low value of WSS leads to an increased possibility of atherosclerotic plaque deposition, thrombosis, or vessel rupture [1,10,15,85].

From Figure 12, it can be seen that the WSS distribution along the analyzed geometry is almost similar during the systolic peak. As for the diastolic phase, however, the WSS distribution varies more markedly, even within the aneurysm, compared to the systolic peak. The maximum average value of WSS at the systolic peak was found to be $5.730$ Pa for the CFD simulation and $4.751$ Pa for the FSI simulation. During the diastolic phase, the corresponding values were $0.876$ Pa for CFD and $0.671$ Pa for FSI. Overall, the rigid-wall CFD approach overestimates the WSS by approximately $21\%$ at systole and $31\%$ at diastole compared to the FSI calculation.

Another parameter investigated is TAWSS, which provides an overview of the entire cardiac cycle of WSS. TAWSS is calculated using the following definition

$$TAWSS={\displaystyle \frac{1}{T}}{\int }_0^T\left|\mathbf{wss}\right|dt$$

(11)

where T is the cardiac period.

The TAWSS distributions obtained in the present study are shown in Figure 13. A high TAWSS value can lead to a greater risk of rupture of the affected walls [86], but reduces the risk of atherosclerosis and thrombus formation [87]. With reference to Figure 13, the TAWSS spatial distribution is similar between CFD and FSI with higher values in the proximal and distal parts of the aneurysm neck and also around the aneurysmal sac where the blood flow impinges on the aortic wall. Calculating the spatial mean of the TAWSS values over the entire surface yields $0.815$ Pa for the CFD case and $0.616$ Pa for the FSI case, i.e., an increase of approximately $32\%$ compared to the FSI simulation. However, compared to the previous overall value of the average spatial TAWSS relative to CFD, greater than in the FSI case, a higher local maximum is obtained in FSI, equal to $4.276$ Pa, compared to the CFD case, equal to $3.223$ Pa, located in the proximal part of the aneurysm. This behavior is due to the fact that deformation can deflect the flow in such a way that it impacts the walls more directly, effectively increasing the TAWSS value in specific areas.

3.4. OSI

The Oscillatory Shear Index (OSI) [88] is a parameter that quantifies the degree of directional oscillation of the WSS and is commonly used to identify areas of the artery subject to oscillatory flow, as defined in Equation (12).

$$OSI=0.5\left(1-{\displaystyle \frac{|{\int }_0^T\mathbf{wss}dt|}{{\int }_0^T\left|\mathbf{wss}\right|dt}}\right)$$

(12)

A value of $0.5$ corresponds to a purely oscillatory flow. With reference to Figure 14, it can be seen that OSI is elevated in the lateral and superior walls of the aneurysm, in accordance with the results described in [41,42]. High OSI values can lead to the formation of atherosclerosis and inflammation of the artery [89].

3.5. Wall Displacement

Figure 15 represents the displacement module on the surface and through the visualization of the vector itself on a slice, showing that the maximum displacement occurs at the systolic peak in the distal neck of the aneurysm. The maximum absolute displacement is $5.60$ mm. Considering section $S_1$ across the aneurysm (cf. Figure 3), the difference between the minimum and maximum equivalent diameter is approximately ≃1.54 mm, corresponding to an increase of $3.59\%$ between the systolic peak ($2.17$ s) and late diastole ($2.90$ s). The maximum displacement does not occur at the instant of maximum pressure but approximately at the systolic peak. This behavior is attributed to the transient nature of the flow and the associated inertial effects. In fact, as shown in Figure 9, the velocity in the outlet region of the aneurysm is higher at the systolic peak than at the pressure peak. The literature on DTAA wall displacement is currently limited; nevertheless, the present results are consistent in order of magnitude with those reported in existing studies [41,42].

3.6. Von Mises

Von Mises Stress (VMS), ${\sigma }_{vms}$, represents the equivalent stress and it is used as a scalar measure of the stress state in the aortic wall. Its definition is given by

$${\sigma }_{vms}=\sqrt{{\displaystyle \frac{{({\sigma }_I-{\sigma }_{II})}^2+{({\sigma }_{II}-{\sigma }_{III})}^2+{({\sigma }_I-{\sigma }_{III})}^2}{2}}}$$

(13)

where ${\sigma }_i$ with $i=I,II,III$ denote the principal stresses of the Cauchy stress tensor. Figure 16 shows the VMS distribution over the entire geometry and within the aneurysm. The maximum value, equal to $0.235$ MPa, is observed in the region proximal to the aneurysm neck rather than within the DTA region. The highest stress levels occur in regions characterized by sharp geometric curvature, where stress concentrations are expected. As anticipated, the VMS remains well below reported rupture thresholds, according to experimental studies ranging from 0.8–2.3 MPa [90,91,92], and therefore does not indicate a risk of aneurysm rupture for the case study examined. It should also be noted that the aneurysmal region is subject to the greatest tensile loading due to the locally higher pressure levels (Figure 10).

3.7. Summary of Results

For further clarity, the main results regarding the comparison of hemodynamic parameters during the systolic peak are summarized in

Table 4. Peak systolic haemodynamic parameters. The difference, expressed as a percentage, is relative to the FSI case. $S_3$ refers to section shown in Figure 3.

Haemodynamic Parameters	CFD	FSI	Difference
Velocity $(S_3)\left[{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}\right]$	$1.204$	$1.045$	$15\%$
Pressure [Pa]	14,797	14,899	$-0.68\%$
WSS [Pa]	$5.730$	$4.751$	$21\%$

and

Table 5. Diastolic haemodynamic parameters. The difference, expressed as a percentage, is relative to the FSI case. $S_3$ refers to section shown in Figure 3.

Haemodynamic Parameters	CFD	FSI	Difference
Velocity $(S_3)\left[{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}\right]$	$0.472$	$0.365$	$29\%$
Pressure [Pa]	10,594	10,464	$1.24\%$
WSS [Pa]	$0.876$	$0.671$	$31\%$

.

The TAWSS parameter is a parameter computed within the cardiac cycle; in our case, the CFD calculation yields a value of $0.815$ Pa, whereas the FSI calculation yields a value of $0.616$ Pa, representing a $32\%$ increase compared to the FSI calculation.

Similarly to the hemodynamic parameters, the main results concerning the structural domain are summarized in

Table 6. FSI simulation structural parameter values. $S_1$ refers to section shown in Figure 3.

Structural Parameters	Value
Max Displacement [mm]	$5.60$
Max diameter difference ($S_1$) [mm]	$1.54$
Max VMS [MPa]	$0.235$

.

For the sake of completeness,

Table 7. Comparison of haemodynamic and structural parameters at systolic peak.

Haemodynamic and Structural Parameters	Present Study	Duca et al. [41]	Difference
${\displaystyle \frac{Velocit{y}_{DTA}}{Velocit{y}_{sac}}}$	$1.7$	$2.3$	$-26\%$
${\displaystyle \frac{Pressur{e}_{DTA}}{Pressur{e}_{sac}}}$	$1.017$	$1.016$	$0.1\%$
Max Displacement [mm]	$5.60$	$6.11$	$-8.35\%$
Max VMS [MPa]	$0.235$	$0.210$	$11\%$

present the values obtained in this study, along with the most recent paper on the DTAA case [41]. For the hemodynamic parameters, in order to compensate for differences in geometry, we use non-dimensional values: the ratio of the maximum velocity/pressure calculated within the aneurysm sac to the maximum velocity/pressure recorded in the DTA section. For structural quantities, instead, we used absolute values.

As can be seen, the velocity ratio shows a $26\%$ difference, but this is primarily due to the geometric difference at the aneurysm neck. For pressure, the difference is essentially zero, since pressure is dominated by the boundary conditions, supporting the accuracy of our simulation.

Table 7 shows also the structural values for the two cases. As can be seen, values for maximum displacement and VMS are of the same order of magnitude, with small deviations despite the different geometries since these quantities are primarily driven by pressure, as highlighted earlier.

4. Conclusions

In the present work, a comparison between rigid-wall CFD and FSI simulations of a DTAA is presented. Pulsatile flow was imposed as a boundary condition at the inlet, while a three-element Windkessel model, implemented in OpenFOAM, was applied as pressure boundary to the carotid and subclavian arteries; a prescribed pressure waveform was applied at the main outlet. Blood flow was modeled as laminar with the incompressible fluid assumption, and the arterial tissue was represented using a Neo-Hookean material model. The main findings can be summarized as follows:

• Higher velocity values were observed during both the systolic and diastolic phases in the CFD simulations compared with the FSI results.
• The pressure distribution is slightly higher in the FSI simulation than in the CFD case during the systolic peak, and, to the best of our knowledge, such aspect is not reported and discussed in the literature. During diastolic phase, the pressure is higher in the CFD simulation as expected and already reported.
• The WSS reflects the trends observed for velocity and pressure, with an increase of $21\%$ and $31\%$ for the CFD simulations compared to the FSI results during the systolic and diastolic phases, respectively.
• The spatial distribution of TAWSS is qualitatively similar between the CFD and FSI simulations; however, the CFD approach overestimates its magnitude by $32\%$ compared to the FSI calculation.
• The OSI reveals marked qualitative differences within the aneurysmal sac, highlighting the influence of wall compliance on local flow oscillations and shear reversal patterns that are not captured by rigid-wall CFD simulations.
• In the solid domain, the maximum displacement of the arterial tissues occurs in the distal region of the aneurysm neck, while the maximum variation in aneurysm diameter between systole and diastole remains below $5\%$. The VMS values are well below the material rupture limits.

The results highlight a significant sensitivity of pressure, wall deformation and stress distributions to the assumed mechanical properties of the arterial wall. The haemodynamic parameters described above, such as WSS and TAWSS, show significant differences between the CFD and FSI models due to the compliance not captured by the rigid-wall model. The aneurysmal sac is not a static tube but acts as a reservoir for blood flow. Thanks to its compliance, the wall expands during systole, storing blood volume and releasing it during diastole, a buffering effect that attenuates mechanical loads. This function is intrinsically related to the shape and volume of the bulge: the larger the aneurysm, the more the rigid model fails to capture this physics. Furthermore, CFD calculations alone do not allow for the analysis of wall deformation and associated stresses. Although no direct clinical validation is provided, these findings suggest that neglecting wall deformation may lead to non-negligible discrepancies in the estimation of haemodynamic quantities in DTAA, potentially affecting their interpretation in a clinical context.

Although the use of a Newtonian model to simulate the blood flow is acceptable, a non-Newtonian model could be used to obtain more realistic results, particularly within the aneurysm sac. In the solid domain a uniform wall thickness and a constant Young’s modulus were adopted in the study, the nature of the case analyzed suggests that spatially varying thickness and elastic properties may be required for further improving the realism of the simulations. Future developments will also take into account not only the inclusion of non-Newtonian fluids, but also the effects of external pressure and the forces arising from the surrounding and constraints on displacement in the common and subclavian arteries.