Fluid–structure interaction analysis of pulsatile flow in arterial aneurysms with physics-informed neural networks and computational fluid dynamics Open Access

Cardiovascular mechanics involve studying the anatomy of the heart and vessels, histology of cardiovascular tissue, and the structure of tissue constituents. Cardiovascular disease remains a pressing clinical concern, impacting ∼92.1 million adults in the United States. Projections indicate that by 2030, nearly 43.9% of the population will be affected by some form of this disease, underscoring the urgency of addressing this public health challenge.¹ 

The intricate nature of blood flow within the cardiovascular system often renders analytical solutions impractical.² Computational fluid dynamics (CFD) methods have evolved since the 1930s, with three-dimensional (3D) CFD methods emerging in the 1960s. CFD has become a popular tool in understanding blood flows in the cardiovascular system, with numerous studies focusing on various aspects such as valve prostheses, arterial bifurcations, and wall shear stress (WSS) distribution. CFD and additive manufacturing for treatment have been discussed in Ref. 1. Vignon-Clementel et al.³ had modeled blood flow and pressure in major arteries with 3D finite elements. The study highlights how different outflow boundary conditions significantly affect flow rate and pressure distributions. In a similar studies, Kim et al.⁴ integrated a lumped parameter heart model as an inflow condition in 3D simulations to accurately capture aortic flow and pressure dynamics, adapting inlet and outlet conditions based on valve status. Further studies have examined the role of local vascular geometries in influencing hemodynamics. Kumar et al.⁵ demonstrated that aneurysm dilation diameter and neck angle play a crucial role in determining local hemodynamic parameters, with saccular aneurysms exhibiting complex streamline patterns and low WSS. The influence of blood rheology has also been investigated, with comparative studies evaluating Newtonian and non-Newtonian models such as Power law, Carreau, Cross, and Herschel Bulkley in simulations of abdominal aortic aneurysm (AAA) and stenosis.⁶ Additionally, pulsatile hemodynamics in patient-specific models of the human abdominal aorta, considering conditions like AAA and right internal iliac stenosis (RIIAS), have been analyzed using different rheological approaches.⁷ Recent advancements in CFD applications have focused on patient-specific modeling for coronary hemodynamics, aiming to improve the diagnosis and understanding of coronary artery disease (CAD). These models utilize a combination of 0D–3D approaches and lumped parameter networks to simulate blood flow and pressure distributions within coronary arteries.⁸ The reader is directed to Refs. 9–11 for further discussion on CFD.

Aorta is the main artery that carries oxygenated blood and transport it into the organs and tissues. Aortic aneurysms (AAs) are characterized by an enlargement of the aorta to 1.5 times or more its normal size.¹² Risk factors for AA include male gender, high blood pressure, advanced age, cigarette smoking, and atherosclerotic disease.¹³ Duprey et al.¹⁴ suggested that the rupture risk is strongly correlated with the physiological elastic modulus, irrespective of age, diameter, or aortic valve phenotype. Burris and Hope¹⁵ discussed 4D flow MRI that enabled the detection of aortic blood flow abnormalities, providing new quantitative markers like flow displacement and wall shear stress that reveal the impact of valve-related abnormalities and the energetic burden in aortic disease. Zorrilla and Soudah¹⁶ presented a cut finite element method (FEM) technique for the efficient CFD simulation of patient-specific aortic dissections. Another paper by Zorrilla et al.¹⁷ discussed a fluid–structure interaction (FSI) framework designed for the robust and efficient simulation of strongly coupled problems involving arbitrarily large displacements and rotations.

Arterial tortuosity, which refers to deviations from a linear path in blood vessels, plays a significant role in influencing hemodynamics and vascular health and is critical to understanding various cardiovascular disorders. Recent CFD studies have explored the effects of tortuosity on arterial health. Ebrahim and Fallah¹⁸ examined how bending tortuosity affects plaque stresses, while Buradi and Mahalingam¹⁹ created idealized models of tortuous arteries, incorporating factors such as curvature radius, bend angle, and inter-bend distance. They also assessed hemodynamic parameters including wall sear stress (WSS), wall shear stress gradient (WSSG), and oscillatory shear index to evaluate the risk of atherosclerosis progression. Their results indicated that areas of the artery wall with higher relative residence duration, significant time-averaged WSS gradient, lower oscillatory shear index, and increased endothelial cell activation potential were more prone to plaque development and rupture. Nagargoje et al.²⁰ studied the internal carotid artery (ICA) siphon in multiple ICA bifurcation aneurysm models with various bend shapes (C-, U-, S-, and helical-shaped) to assess the initiation, progression, and rupture of aneurysms. Their findings showed that features of the parent artery, such as its shape, tortuosity, curvature, and number of turns, significantly influenced the associated hemodynamics. Similarly, Qiu et al.²¹ investigated the hemodynamics of patient-specific abdominal aortic aneurysms with iliac artery tortuosity and found that WSS in tortuous aortas was three times higher than in normal aortas. Li et al.²² also analyzed descending aorta models with varying degrees of tortuosity, focusing on hemodynamic parameters. Kliś et al.²³ utilized CFD to study the impact of tortuosity on hemodynamics in brain aneurysms, specifically using basilar artery models.

Hemodynamic forces significantly influence the development and progression of vascular diseases, with FSI playing a critical role in evaluating the biomechanics of blood vessels and prosthetic devices.¹¹ FSI occurs when fluid flow imparts forces onto the structural surfaces of biological tissues or devices, creating a dynamic interplay between the fluid and structural components. This section explores various computational and experimental methodologies that have advanced our understanding of FSI and its implications in vascular disease modeling. Hsu et al.²⁴ investigated FSI in bio-prosthetic heart valves, emphasizing the dynamic interaction between the fluid and structural elements. Their findings demonstrated the importance of accurately capturing these interactions to improve prosthetic valve designs. Bavo et al.²⁵ compared Arbitrary Lagrangian–Eulerian (ALE) and Eulerian FSI approaches in aortic valve simulations. Their study highlighted the superior accuracy of the ALE approach at the solid–fluid interface, making it more effective for high-fidelity simulations. In the context of aneurysm research, Philip et al.²⁶ analyzed fusiform aneurysms by considering the shape index (DHr) and its relationship to von Mises stress. They discovered that aneurysms with low DHr values exhibited heightened von Mises stress and significant deformation, whereas regions with high DHr values showed reduced stress and minimal deformation. Scotti and Finol²⁷ further examined physiological parameters' effects on FSI and computational solid stress (CSS) in patient-specific abdominal aortic aneurysm (AAA) models. Their findings revealed that while CSS methods can predict wall stress within 5% accuracy for specific morphologies, they may underestimate maximum wall stress by up to 25%. Salman et al.²⁸ provided a comprehensive overview of computational modeling and experimental setups for studying AAA rupture mechanics. Their work identified elevated WSS and structural wall stress as critical indicators of rupture risk. Expanding on this, Wang et al.²⁹ introduced a finite element-based method to simulate tissue degradation in aneurysmal walls and its effects on intra-aneurysm hemodynamics. The study revealed that tissue degradation reduces wall stiffness, increases strain, and results in spatially heterogeneous changes in WSS, with elevated WSS in flow impingement areas and reduced WSS in others. These findings collectively underscore the complex role of FSI and WSS in aneurysm progression and rupture.

Physics-informed neural networks (PINNs) have emerged as a transformative framework for addressing forward and inverse problems involving nonlinear partial differential equations (PDEs), offering substantial computational advantages over traditional methods. Introduced by Raissi et al.,³⁰ PINNs embed physical laws directly into neural networks, reducing reliance on extensive datasets and enabling efficient solutions to PDEs. By eliminating the need for complete CFD simulations when boundary conditions or geometries change, PINNs act as rapid surrogates, facilitating iterative parameter sweeps, sensitivity analyses, and real-time clinical decision-making. Their potential in hemodynamics is particularly evident in applications such as Marfan syndrome (MS) aortic aneurysms, where they expedite assessments of WSS and von Mises stress while preserving computational efficiency. Recent advancements, including self-adaptive PINNs introduced by McClenny and Braga-Neto,³⁵ have further refined this framework by incorporating soft attention mechanisms to focus on challenging regions during training. The role of activation functions has also been explored, with Ramachandran et al.³⁶ identifying effective activation strategies and He et al.³⁷ demonstrating the superior performance of rectifiers. Tools like PyTorch, developed by Paszke et al.,³⁸ have facilitated the implementation of PINNs, while optimization techniques such as decoupled weight decay regularization³⁹ and mixed precision training⁴⁰ have improved training stability and accelerated computations. Wang et al.⁴¹ critically evaluated the commonly used $L2$ physics-informed loss, offering insights for designing more effective loss functions. In hemodynamic modeling, Arzani et al.⁴² applied PINNs to extract accurate WSS data from velocity measurements, overcoming the challenges associated with traditional CFD approaches. Beyond PINNs, machine learning as a whole has revolutionized cardiovascular research, enabling the analysis of large datasets to capture complex interactions between vessel geometry, blood flow dynamics, and wall stress. Habibi et al.⁴³ utilized machine learning to predict aneurysm rupture risk using data from over 18 000 participants, while Al'Aref et al.⁴⁴ highlighted the transformative role of machine learning in cardiac imaging. Comprehensive reviews by Austin⁴⁵ and Brunton⁴⁶ further emphasize the growing impact of machine learning in fluid dynamics and cardiovascular studies. Collectively, these advancements underscore the synergy between PINNs, machine learning, and CFD, streamlining workflows, enhancing computational efficiency, and offering new opportunities for patient-specific hemodynamic analyses and clinical applications.

Marfan syndrome is a genetic disorder that affects connective tissue, providing structural support to various body parts, including the heart, blood vessels, eyes, bones, and skin. The condition is caused by a mutation in the fibrillin-1 (FBN1) gene, which disrupts the formation of elastic fibers in connective tissue. This leads to weakened connective tissue, particularly in the cardiovascular system, where the aorta is most affected. The weakened aortic wall becomes more susceptible to dilation, increasing the risk of aortic aneurysms. Aneurysms in Marfan syndrome patients most commonly occur in the ascending aorta or aortic root, and their progressive enlargement can lead to life-threatening conditions, such as aortic dissection. Several studies have examined the relationship between Marfan syndrome and aortic aneurysms. Lenz et al.⁴⁷ conducted a hierarchical cluster analysis using aortic 4D flow MRI to investigate the hemodynamics of the aorta in Marfan syndrome patients. van Andel et al.⁴⁸ studied abnormal aortic hemodynamics and found that WSS in the proximal descending aorta is linked to known risk factors for aortic dissection in Marfan syndrome patients. Ma et al.⁴⁹ discussed the clinical applications of cardiovascular magnetic resonance (CMR) in Marfan syndrome, highlighting the importance of monitoring aortic dilation and assessing the risks of aneurysm formation and dissection.

This study aims to improve the understanding of hemodynamics in Marfan syndrome aneurysms, which is crucial for assessing rupture risk and supporting clinical decision-making. We integrate CFD with PINNs to analyze WSS and von Mises stress in both healthy and aneurysmal aortic models. The analysis includes six models, four representing aneurysms caused by Marfan syndrome and two representing healthy cases, sourced from the Vascular Model Repository. Simulations were conducted using ANSYS Fluent software to visualize patient-specific blood flow under pulsatile flow conditions. To replicate realistic physiological conditions, FSI framework was applied along with a pulsatile inlet velocity profile, capturing the time-dependent behavior of blood flow. Additionally, the shear stress transport (SST) k– $ω$ turbulent transitional model was utilized to enhance the accuracy of transitional flow simulations. This comprehensive approach highlights the differences in hemodynamic patterns between healthy and diseased aortas, while demonstrating the potential of PINNs as a computationally efficient alternative for hemodynamic modeling, ultimately paving the way for faster clinical evaluations and improved patient care.

In this study, we analyzed four models of Marfan syndrome aneurysm and two models from healthy individuals. These models were sourced from the open-source Vascular Model Repository (http://www.vascularmodel.org).⁵⁰ The reader is referred to the Vascular Model resources for details on patient selection, imaging data acquisition, and 3 D model generations. The models were initially converted into solid form using Ansys SpaceClaim, followed by CFD analysis using Ansys Fluent to assess WSS and other hemodynamic factors. Detailed information regarding all models is provided in Table I and 3D models are depicted in Fig. 1.

Blood, composed of cells suspended in plasma, demonstrates non-Newtonian behavior, particularly notable in vessels with diameters smaller than 1 mm.⁵¹ In certain pathological and transient conditions, such as artery dilatation and bifurcations, shear rates may decrease below $100 s−1$⁠,⁵² underscoring the potential importance of non-Newtonian processes.^53–55 Conversely, in larger capillaries where shear rates typically exceed $100 s−1$⁵² and with diameters exceeding 0.3 mm, blood can be reliably represented as a Newtonian fluid.^56–60 For the purposes of this study, a Newtonian fluid model was selected due to its practicality and relevance in modeling arterial blood flow. This model takes into account blood density of 1050 kg/m³ and viscosity of 0.0035 Pa s.⁶¹ 

Here, $xi$ denotes the spatial coordinates, t is the time, $Ui$ is the averaged velocity, $ui$ is the fluctuating velocity, p is the averaged pressure, and $ρ$ is the fluid density. To close the RANS equations, the Reynolds stress term, $−uiuj¯$⁠, must be modeled. The transition (⁠ $γ$-Re $θ$⁠) SST model, which is a four-equation eddy viscosity model, is employed for turbulence modeling. This model incorporates transport equations for four variables: the turbulent kinetic energy (k), the specific dissipation rate (⁠ $ω$⁠), intermittency (⁠ $γ$⁠), and the transition momentum thickness Reynolds number (⁠ $Rẽθ$⁠).

In these equations, $Gk*$ and $Gω$ represent the production terms for k and $ω$⁠, respectively. The terms $Γk$ and $Γω$ denote the effective diffusivities of k and $ω$⁠, while $Yk$ and $Yω$ indicate their dissipation. The cross-diffusion term is represented by $Dω$⁠, and $Sk$ and $Sω$ are user-defined source terms.

To represent the physiological fixation of the artery, fixed boundary conditions were applied at both the inlet and outlet. The FSI interface was applied to the inner surface of the arterial wall to allow for bidirectional interaction between the fluid and structural domains.

The governing equations were solved using the ANSYS Fluent commercial solver, which employs the finite volume method and a pressure-based approach to address the non-linear equations governing mass and momentum conservation. To ensure convergence, a residual error threshold of $10−3$ was set. A mesh independence study was conducted under steady-state flow conditions with an inlet velocity of 0.5 m/s, using the SST k– $ω$ turbulence model to determine the optimal mesh size, which was found to be between 0.7 and 0.9 mm. This range was selected based on tests showing that the variation in WSS between the finer and finest meshes remained below 2%, depending on the specific model analyzed. The results of the mesh independence analysis for Marfan syndrome cases, particularly at the aneurysm sacs, are presented in Table II. To ensure a balance between computational efficiency and accuracy, a finer mesh was chosen for all cases.

Additionally, a time step independence study was conducted by varying time steps to evaluate the impact of time step size on the simulation results, particularly the computed WSS values. A time step of 0.001 s was chosen after observing negligible differences in WSS values. Using this time step, the simulation was run for a duration of 2 s, resulting in a total of 2000 iterations.

High-fidelity CFD simulations, as outlined in Sec. II F, offer precise insights into arterial hemodynamics and are central to modeling fluid dynamics in complex vascular geometries. However, these simulations can be both computationally expensive and time-consuming, especially when dealing with patient-specific anatomies or conducting numerous parameter sweeps. Such challenges are further compounded by the need for significant domain expertise to refine meshes, tune solvers, and interpret results.

PINNs, first introduced by Raissi et al.,³⁰ have emerged as a powerful alternative that combines the strengths of traditional deep learning with established physics-based modeling. Rather than relying exclusively on data-driven learning, PINNs integrate the governing partial differential equations (PDEs) in this case, the continuity and momentum conservation equations (Sec. II C) directly into the network's loss function. This approach ensures that the neural network is “informed” by the physical laws of fluid motion, guiding it toward realistic flow solutions even when labeled data are sparse or imperfectly measured.

By integrating PDE residuals and boundary conditions into the training process, PINNs inherently enforce the fundamental laws of fluid dynamics, yielding physically consistent solutions even with limited labeled data. This physics-based integration minimizes overfitting and eliminates the need for conventional mesh discretization, enabling the network to be evaluated at arbitrary spatial and temporal resolutions, and as a result, they can effectively complement or even partially replace traditional CFD simulations. Moreover, the outputs of the PINNs can be used to generate high-resolution flow-field predictions that exceed the original CFD. Consequently, once trained, PINNs provide rapid inference and are highly adaptable to new boundary conditions or geometrical changes with minimal retraining, making them an efficient, mesh-free alternative to traditional CFD simulations.³¹ 

Furthermore, the mesh-free framework of PINNs eliminates the need for domain discretization, thus simplifying simulation workflows and reducing computational overhead. This attribute is particularly beneficial for complex geometries such as arterial aneurysms, where conventional CFD methods often face challenges during mesh generation.^32,33 Techniques such as adaptive weighted loss functions further improve prediction accuracy and generalization capabilities, making PINNs a versatile tool for a wide range of fluid dynamics applications.³⁴ 

Although PINNs provide numerous benefits, ensuring robust and accurate simulations across various physiological conditions remains a challenge. Incorporating clinical data and further refining neural network architectures could significantly broaden the utility of PINNs in real-world cardiovascular diagnostics and treatment planning.

In this study, we combine PINNs with traditional CFD data to estimate core hemodynamic variables such as pressure, velocity components, and WSS across healthy and Marfan syndrome aortic geometries presented in Sec. II A. This unified approach enables direct comparisons of velocity patterns, shear forces, and other critical flow indicators between non-pathological and aneurysmal models, providing valuable information for understanding the progression of vascular disease. Sections III A–III D detail the formulation, architecture, and training strategy of the PINNs used in this study.

Modeling pulsatile flow in arterial segments requires the enforcement of continuity and momentum conservation equations to capture essential hemodynamic effects. In this work, blood flow is modeled using Reynolds-averaged Navier–Stokes (RANS) equations (see Secs. II B and II C), which describe the flow behavior in the aortic geometries. Let $u=(u,v,w)$ denote the velocity field, p the pressure, $ρ$ the fluid density, and $μ$ the dynamic viscosity. The continuity and momentum conservation equations are combined to form the Navier–Stokes system, which governs the evolution of the velocity field and pressure in response to external forces and viscous effects.

The associated boundary conditions involve a no-slip condition on the arterial walls $(u=0 on ∂Ωwall)$⁠, zero relative pressure at the outlet, and a pulsatile inlet velocity on $∂Ωinlet$ (Sec. II B). These boundary and inlet constraints are incorporated into the physics-informed neural network through penalty terms in the loss function, thereby enforcing consistency between the network's predictions and the specified physical conditions. Temporal dependence is explicitly modeled to reflect the transient nature of blood flow.

Neural network architecture plays a crucial role in the performance and generalization capabilities of PINNs. In this study, we train separate PINNs for pressure p, velocity components $(u,v,w)$⁠, and the three WSS components $(τx,τy,τz)$⁠, all time dependent $(t)$⁠. Each network is a feed-forward, fully connected architecture with $L=10$ hidden layers and $N=64$ neurons per layer. The Swish activation function³⁶ is employed to promote smooth gradients. Batch normalization is applied to each layer to stabilize training, and all weights are initialized using Kaiming normal³⁷ to ensure stable gradient flow.

The self-adaptive loss weighting scheme is a key component of the PINN training process, enabling the network to dynamically adjust the relative importance of different loss components based on their contributions to total loss. This mechanism prevents any single loss term from dominating, thereby ensuring that the network effectively balances the physics residuals, boundary conditions, inlet constraints, and supervised data.

Balancing multiple loss components is a fundamental challenge in training PINNs, particularly in multi-physics scenarios where various physical phenomena and boundary conditions must be simultaneously satisfied.³⁵ In our study, the primary loss components include the physics residuals derived from the governing partial differential equations (PDEs), boundary conditions, inlet conditions, and supervised data from CFD simulations. Improper weighting of these components can lead to suboptimal training outcomes, such as overfitting to certain loss terms or failure to adequately satisfy physical laws.

During training, the network weighted $θ$ and the log-weights $log λi$ were updated simultaneously using stochastic gradient descent. The loss components were computed for each minibatch, and the total loss was evaluated on the basis of the current predictions and the CFD reference data. The total loss gradients with respect to the network weights and the log weights were then backpropagated to update the parameters. A StepLR scheduler was used to adjust the learning rates for the network weights and log weights every N epoch, ensuring stable convergence and preventing oscillations during training. The self-adaptive weighting mechanism enables the network to effectively balance the physics residuals, boundary conditions, inlet constraints, and supervised data, leading to robust and accurate predictions of arterial flow dynamics.

The experimental data were derived from CFD simulations of both healthy and aneurysmal aortic geometries, with measurements acquired during the systolic and diastolic phases. Each dataset comprises spatial coordinates (X, Y, Z), time points, pressure values, velocity components (u, v, w), and wall shear stress (WSS) components. Prior to model training, the raw data underwent a comprehensive preprocessing pipeline. Initially, the data were cleaned by removing incomplete records and verifying the physical consistency of the measurements, resulting, for example, in the 0021 diastolic aneurysm dataset containing approximately 11 005 spatial points per time step. Subsequently, all input characteristics and target variables were normalized via Min–Max scaling; spatial coordinates were scaled to the range $[−1,1]$⁠, the time points were normalized over the cardiac cycle and each of the physical variables (pressure, velocity, and WSS components) was independently scaled to ensure robust training. To enforce the no-slip condition at the vessel walls, points were identified using a velocity magnitude threshold (⁠ $|u|, |v|, |w|<10−5$⁠). The processed data were then organized by healthy cases (0024 and 0142) and aneurysmal cases (0021, 0022, 0023, and 0025) with uniform temporal sampling throughout the cardiac cycle. Finally, rigorous quality control measures were applied, including consistency checks in velocity and pressure fields, validation of WSS components against analytical solutions, verification of mass conservation at the inlet and outlet boundaries, and confirmation of temporal continuity across cardiac cycles.

The PINN framework was implemented using the PyTorch library,³⁸ leveraging its automatic differentiation capabilities for efficient computation of PDE residuals and trained on a high-performance computing cluster with NVIDIA Rtx8000.

The input spatiotemporal data points, $(xj,yj,zj,tj)$ which are geometric were randomly sampled from the aortic geometries and over the cardiac cycle. The geometric and variables data were normalized using Min–Max scaling to facilitate efficient training. The PINNs were trained using the AdamW optimizer³⁹ with an initial learning rate of $1×10−4$ and momentum parameters $β=(0.9,0.999)$⁠. The weight decay was set to $1×10−4$ to prevent overfitting. A StepLR scheduler reduced the learning rate by a factor of $γ=0.9$ every 200 epochs to facilitate finer convergence.

Training was conducted using mixed-precision arithmetic via PyTorch's “torch.cuda.amp” to enhance computational efficiency and reduce memory consumption without sacrificing model accuracy.⁴⁰ The training process iterated over a maximum of 1000 epochs, with early stopping implemented to halt training if the validation loss did not decrease for 5 consecutive epochs. Gradient clipping was applied to prevent exploding gradients; the gradients are clipped to a maximum norm of 1.0, ensuring training stability (Fig. 4).

In the current study, WSS distributions and von Misses stress were analyzed for four models of Marfan syndrome aneurysm and compared with a healthy aorta model during both systolic and diastolic phases. The details of these models are presented in Table I and 3D models are depicted in Fig. 1.

Figures 5(a) and 5(b) represent the wall shear stress contours for MS₁ aneurysm model, Figs. 5(c) and 5(d) represent a MS₂ model, Figs. 5(e) and 5(f) correspond to a MS₃ model, and Figs. 5(g) and 5(h) display the WSS distributions for the MS₄ aneurysm model. Figures 5(i) and 5(j), and 5(k) and 5(l), illustrate the WSS distributions for healthy aortic models (H₁ and H₂), providing a baseline for comparison.

In Marfan syndrome aneurysm models, the WSS distributions [subfigures (a), (c), (e), and (g) of Fig. 5] exhibit significant spatial heterogeneity, particularly in regions of aneurysmal expansion. During the systolic phase, WSS values peak in areas of sharp curvature and near the walls of the aneurysm (proximal areas), with maximum values reaching as high as 15.53 Pa in some regions. This trend is consistent across all four cases, with WSS consistently higher at the posterior, lateral, and superior walls compared to the anterior wall. Notably, the highest value is observed on the superior side of case MS₁, indicating the highest rupture risk on that side compared to other cases. High WSS values during systole can exacerbate endothelial damage, weakening the vessel walls and increasing the risk of rupture. While WSS is slightly elevated in non-aneurysmal regions compared to within the aneurysmal sacs, the overall distribution remains highly non-uniform, with lower values within the aneurysms and higher values in areas like the carotid and subclavian arteries, which can contribute to thrombus formation. The elevated WSS in the carotid and subclavian arteries can be attributed to their anatomical and hemodynamic features. These arteries branch from the aorta at regions of considerable curvature, leading to disturbances in blood flow, with flow acceleration as it enters the smaller-diameter arteries, further increasing WSS. In contrast, the aneurysmal regions, with their larger diameters and complex curvature, slow down the flow and reduce WSS. The descending thoracic aorta experiences tortuosity and centrifugal forces, which push the flow outward, leading to higher WSS on the posterior, lateral, and superior walls, and lower WSS on the anterior side. Tortuosity plays a significant role in this dynamics, as the flow experiences centrifugal forces that push the blood against the outer curve of the aneurysm, resulting in an increase in shear stress. These centrifugal forces cause flow acceleration on the anterior side and deceleration on the posterior side, creating a velocity difference between the two, which leads to lower WSS on the anterior side. This difference in WSS contributes to endothelial damage and may play a role in the progression of aneurysm-related complications. These hemodynamic changes lead to non-uniform WSS distributions, with lower WSS in aneurysmal regions promoting thrombus formation.

During the diastolic phase [subfigures (b), (d), (f), and (h) of Fig. 5], WSS values decrease but remain non-uniform, with some regions continuing to experience moderate WSS that could induce chronic vascular stress. These values range from 2.83 to 5.84 Pa, depending on the specific geometry of the aneurysm and possibly the age of the patient, suggesting that even during diastole, mechanical forces still impose considerable stress on the aneurysmal walls. This persistent stress could contribute to long-term weakening of the vessel, increasing the risk of rupture. Notably, there are marked differences in WSS between the anterior, posterior, lateral, and superior walls of the aneurysms, reflecting the complex flow patterns that result from the unique shapes of each aneurysm.

In contrast, the healthy aorta models [subfigures (i)–(l) of Fig. 5] exhibit a much more uniform distribution of WSS across the vessel surface. During the systolic phase, the maximum WSS reaches ∼143.23 Pa; however, this elevated WSS is primarily concentrated at the carotid and subclavian arteries. These areas experience higher shear stress due to their anatomical locations, where the vessels branch off from the aortic arch, creating regions of flow acceleration and turbulence. Aside from these arterial branches, the WSS remains below 40 Pa across the majority of the aorta, which demonstrates a more consistent and homogeneous distribution of shear stress compared to the aneurysmal models. The uniformity of WSS in the healthy models suggests that these vessels are well-equipped to handle the pulsatile nature of blood flow without the extreme localized stresses observed in diseased cases. This steady and even distribution of mechanical stress likely contributes to the structural integrity of the aortic walls, reducing the likelihood of pathological changes such as aneurysm formation. During the diastolic phase [Figs. 5(j) and 5(l)], WSS values decrease substantially, dropping below 6 Pa in most areas. This further highlights the adaptability of healthy vessels to varying hemodynamic conditions, as they can effectively reduce shear stress during periods of lower blood flow. The absence of significant stress concentrations during both systole and diastole reinforces the stability and resilience of healthy aortic tissue. In contrast to the highly variable and reduced WSS observed in aneurysmal regions, the uniformity of WSS levels in healthy aortas reduce the risk of endothelial damage and, consequently, the development of vascular pathologies such as aneurysms or rupture.

FSI plays a critical role in accurately modeling the mechanical behavior of arterial walls under pulsatile blood flow conditions. By coupling the fluid dynamics of blood flow with the deformation of the vessel walls, FSI provides insights into the stress distribution within the arterial walls, which is crucial for understanding the risk of aneurysm rupture and other vascular complications. The von Mises stress, which arises from FSI, is a key indicator of the material response of the vessel walls under varying stress conditions. It helps identify areas where the arterial wall is most likely to experience failure due to excessive stress, particularly under the dynamic loading conditions of arterial blood flow. Figure 6 illustrates the von Mises stress (Pa) distributions for cases MS₁–MS₄. In subfigure (a), which represents case MS₁, the von Mises stress is observed to be elevated at the distal end of the anterior side and on the superior side. This suggests that these areas of high stress may be prone to structural weakening, with the potential for aneurysm rupture being higher in these regions. Subfigure (b) shows case MS₂, where the von Mises stress is notably higher at the proximal end of the posterior side and on the superior side, indicating that these regions experience significant stress concentration. This could imply a higher rupture risk at the proximal posterior and superior regions in this specific case. In subfigure (c), which represents case MS₃, the von Mises stress is again observed to be higher at the distal end of the anterior side, suggesting that similar to case MS₁, the anterior side of this aneurysm model faces elevated stress in the distal regions, which could contribute to potential rupture or other complications. Finally, subfigure (d) illustrates case MS₄, where the von Mises stress is predominantly high on the superior side. This indicates that, in this case, the superior side is under the greatest stress, increasing the likelihood of rupture in this region. When comparing these findings across all four cases, it is evident that regions with higher von Mises stress are more susceptible to rupture. Among all the cases, case MS₁ exhibits the highest von Mises stress on the superior side, suggesting that the rupture risk is particularly high in this region for this case. This is crucial for understanding the hemodynamic and mechanical factors contributing to aneurysm rupture, as regions experiencing elevated von Mises stress are more likely to experience failure under the dynamic conditions present in the cardiovascular system. Therefore, the superior side of case MS₁ stands out as the most vulnerable to rupture in comparison to the other cases, and addressing the stress concentrations in these areas could be pivotal for preventing aneurysm rupture in clinical settings.

The PINN-based simulations were validated by comparing them with benchmark CFD data for both healthy and Marfan syndrome aortic geometries, demonstrating strong agreement in WSS patterns and pressure distributions, as illustrated in Fig. 7. The PINNs were able to accurately capture hemodynamic factors associated with aneurysm rupture. The self-adaptive loss weighting mechanism effectively balanced the contributions from the physics residuals, boundary conditions, and data-fitting objectives, ensuring that the network learned realistic flow solutions while remaining consistent with the governing Navier–Stokes equations. All numerical experiments and code implementations for the PINN-based aneurysm flow analyses were carried out using Python with PyTorch. Figure 7 illustrates the comparison between the PINN predictions and the CFD reference data for both pressure and WSS. These visual comparisons confirm that the PINN accurately reproduces the key flow characteristics observed in the CFD simulations. Furthermore, the evolution of the loss curves over 1000 training epochs highlights the convergence behavior of the physics, boundary, inlet, and data losses. The steady decline of these loss components reflects the efficacy of the self-adaptive weighting mechanism in guiding the network toward a physically consistent solution.

The mesh-free nature of PINNs³⁰ enables rapid evaluation of flow fields at arbitrary spatial and temporal resolutions, facilitating detailed hemodynamic analyses and personalized treatment planning. Using the self-adaptive loss weighting scheme, the PINNs effectively balance the physics residuals, boundary conditions, and data-fitting objectives, ensuring that the network learns realistic flow solutions while adhering to the governing Navier–Stokes equations. The results presented here demonstrate the potential of PINNs as a powerful tool for simulating pulsatile flow in arterial geometries, with applications ranging from disease diagnosis to treatment planning and optimization.

The findings of this study hold significant clinical relevance, particularly in the early detection and management of aortic aneurysms, with a specific focus on patients with Marfan syndrome. By shedding light on the relationship between WSS distributions and aneurysm progression, this research could pave the way for more targeted diagnostic and therapeutic strategies. Identifying regions within the aneurysm that experience low WSS could assist clinicians in recognizing areas that are at higher risk for thrombus formation and rupture, thereby enabling more precise monitoring and intervention. For example, aneurysmal regions exhibiting significantly lower WSS compared to the carotid and subclavian arteries may benefit from more frequent imaging and tailored treatment plans aimed at reducing the risk of rupture.

Moreover, the study's findings could inform the development of preventive measures, such as surgical interventions or pharmacological treatments, designed to normalize WSS patterns in vulnerable regions. Understanding how variations in WSS contribute to aneurysm growth and rupture could lead to therapies targeting blood flow dynamics to reduce shear stress in high-risk areas, thereby preventing aneurysmal progression.

Despite the valuable insights provided, several limitations must be considered. First, the current study primarily focuses on WSS distributions, but incorporating additional mechanical factors such as vessel wall elasticity, stress–strain relationships, and material properties would provide a more comprehensive understanding of aneurysm behavior. Due to computational limitations, these aspects are not included in the current study but are planned for future research. Furthermore, the results of this study may not fully reflect the complexity of real-world human anatomy. Variations in patient-specific aortic geometry, flow patterns, and material properties may lead to differences in WSS distributions and aneurysmprogression. In addition, the boundary conditions were not patient-specific. Therefore, while the findings are promising, validation with patient-specific data from clinical studies is necessary to confirm their clinical applicability. Finally, while the study offers valuable insights into the role of WSS in aneurysm formation and rupture, it does not account for other factors that may influence aneurysm behavior, such as genetic predispositions, hypertension, and comorbidities. A more holistic approach that considers these additional variables would enhance our understanding of Marfan syndrome and its vascular complications.

In conclusion, this study underscores the critical role of WSS in Marfan syndrome aneurysms and lays the foundation for future research aimed at improving clinical management strategies. Despite the limitations, the findings contribute to the growing body of knowledge regarding the hemodynamic factors influencing aneurysm development, opening avenues for more personalized and effective treatment approaches.

This study provides a comprehensive hemodynamic analysis of Marfan syndrome aneurysms, highlighting the critical roles of WSS and von Mises stress in the progression and rupture risk of aortic aneurysms. The findings underscore significant differences between healthy and diseased aortic models: healthy aortas exhibit uniform WSS distributions, whereas diseased models display highly non-uniform patterns. In aneurysmal regions, reduced WSS was associated with thrombus formation, while elevated WSS in areas such as the carotid and subclavian arteries was linked to geometric and hemodynamic complexities. von Mises stress analysis identified regions of heightened rupture risk, particularly on the superior side of case MS₁, where the highest stress levels were recorded.

By integrating CFD with PINNs, this study demonstrates a computationally efficient and accurate approach for modeling hemodynamics, enabling rapid clinical decision-making. The use of the SST k– $ω$ turbulent transitional model effectively captured transitional flow behaviors, while FSI provided insights into the interaction between blood flow and vessel walls. Although the study employed Newtonian fluid assumptions, the results offer valuable insights into the complex interplay of hemodynamic forces in Marfan syndrome aneurysms. This research emphasizes the potential of advanced computational models to inform diagnostic and therapeutic strategies, paving the way for personalized treatment and better management of vascular diseases.

This work has been supported by the TÜRKİYE Burslari (YTB) scholarship. We extend our heartfelt gratitude to the Turkish government for their invaluable support, which has significantly contributed to the success of this research. Additionally, this work received funding from the Zhejiang Provincial Public Welfare Research Project (Grant Nos. LGC22H180003 and 2023ZL563) and the Science and Technology Development Project of Hangzhou (Grant No. 2021WJCY254).

This study employs publicly available open-source models associated with Marfan syndrome aneurysms obtained from the www.vascularmodel.org website. As these models are publicly accessible, the research does not involve direct experimentation with human participants or animals.

The authors have no conflicts to disclose.

M. Abaid Ur Rehman: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Özgür Ekici: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal). M. Asif Farooq: Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Khayam Butt: Formal analysis (equal); Investigation (equal); Validation (equal); Visualization (equal). Michael Ajao-Olarinoye: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Zhen Wang: Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Validation (equal); Visualization (equal). Haipeng Liu: Formal analysis (supporting); Funding acquisition (lead); Investigation (supporting); Validation (supporting); Visualization (supporting).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.