Despite significant advancements in medical therapy and the availability of both invasive and noninvasive diagnostic tests, cardiovascular disease continues to persist as the leading cause of mortality globally (Maurovich-Horvat et al., 2014; Benvidi and Firoozabadi, 2024). It is widely acknowledged that coronary artery hemodynamics and vessel geometry play a crucial role in the development and diagnosis of coronary artery disease (CAD) (Pinho et al., 2019; Eslami et al., 2020a). Due to the presence of turbulence in the flow, the frictional force exerted on the vessel wall, known as wall shear stress (WSS), can initiate plaque formation at regions of flow separation (Dong et al., 2015; Temov and Sun, 2016).

The left coronary artery and its bifurcations, specifically the left anterior descending (LAD) and left circumflex (LCX), are susceptible to the development of atherosclerosis (Zhang and Dou, 2015; Edrisnia et al., 2024). Additionally, bifurcations pose a heightened risk for restenosis and stent thrombosis (Chaichana et al., 2011; Beier et al., 2016; Freidoonimehr et al., 2021a; Freidoonimehr et al., 2021b). Hence, in clinical practice, it is imperative to assess these regions to identify high-risk patients who may have CAD at an early stage (Hoque et al., 2021). Parameters based on WSS, such as time-averaged wall shear stress (TAWSS) play a fundamental role in the progression of atherosclerosis, particularly in transient studies (Arzani et al., 2021). Studies have shown that a wider bifurcation angle is associated with lower wall shear stress and higher oscillatory shear index (OSI), which can contribute to the stimulation of atherosclerosis (Sun and Chaichana, 2016; Sun and Chaichana, 2017; Liu et al., 2019). Moreover, precise characterization of blood rheology, which involves the application of non-Newtonian models, is particularly crucial in these instances (Pinto and Campos, 2016; Sandeep and Shine, 2021).

Due to the shear thinning behavior of blood in small arteries, CFD is becoming increasingly essential in hemodynamics’ calculations. The accuracy of input data, such as vessel geometry information, appropriate boundary conditions, and material models, significantly influences the reliability and precision of CFD results (Gijsen et al., 2019; Rabbi et al., 2020; Edrisnia and Sharbatdar, 2024). However, the CFD technique is not well-suited for fast-response applications due to the inherent delay and relatively high computational costs involved. As a result, there is a need for an alternative approach to overcome this limitation.

Many state-of-the-art machine learning (ML) techniques, such as Convolutional Neural Networks (CNNs) and Multi-Layer Perceptrons (MLPs), are widely utilized for various applications, including 2D image segmentation and 3D point cloud segmentation (Bello et al., 2020; Hirzi et al., 2022; Sabet et al., 2022). Inspired by these algorithms, there is significant potential for employing them in hemodynamic parameter prediction. However, in scenarios with limited data availability, deep learning models, such as convolutional and recurrent neural networks, often lack robustness and fail to guarantee convergence. Therefore, ensuring access to sufficient and high-quality data is crucial for achieving reliable and accurate results. One promising approach to address data limitations is the use of idealized models in biomedical applications, which can simulate conditions with acceptable agreement to patient-specific cases. These enriched datasets improve algorithm training, enhancing their robustness and performance in real-world applications.

In fluid dynamics applications involving direct calculation of the flow field, ML, and deep learning (DL) approaches have been employed to address computational challenges. These techniques offer promising solutions for expediting computations and enabling fast-response simulations. For instance, Jordanski et al. (Jordanski et al., 2018) have utilized ML techniques to predict the WSS distribution in carotid bifurcation and abdominal aortic aneurysms. Liang et al. (Liang et al., 2017) have deployed a ML approach to establish the connections between shape features and the rupture risk of ascending aortic aneurysms as predicted by finite element analysis (FEA). Furthermore, Liang et al. (Liang et al., 2018; 2020) have developed a DL model, including deep neural networks, to directly estimate the stress distributions in the aorta. Madani et al. (2018) have created data-efficient DL classifiers for cardiology prediction tasks, with a particular emphasis on relevant structures using pipeline-supervised models. Additionally, Madani et al. (2018) have investigated the connection between ML and finite element modeling for WSS prediction of arterial in the context of atherosclerosis. Farnoud et al. (2023) have employed random forest and gradient boosting models for ML analysis of drug delivery to nasal epithelium.

Similar studies are summarized in Table 1. The application of ML or DL to predict hemodynamic parameters, such as Fractional Flow Reserve, FFR, and forecast flow field patterns has remained limited (Itu et al. (2016)). However, there has been an implementation of a reduced-order model, which is highly specialized but has limited applicability. In other investigations, many researchers have employed the Point Net architecture to predict hemodynamic parameters in cardiovascular geometries. Li et al. (2021a) have developed a DL approach to predict the hemodynamics of different cerebral aneurysms before and after the insertion of a flow diverter (FD) stent. Moreover, Li et al. (2021b) have created cardiovascular hemodynamic point datasets and a dual sample channel DL network, capable of analyzing and reproducing the relationship between cardiovascular geometry and internal hemodynamics. The PointNet algorithm, originally proposed for hemodynamic prediction, is also tested in this study to compare its performance with the proposed model, as shown in Table 1.

Studies on PointNet consistently highlight its strength in capturing global geometric variations but reveal its limitations in addressing local variations. To overcome this, researchers often train separate PointNet models for pre-treatment and post-treatment simulations. In contrast, Su et al. (2020) proposed an alternative approach using idealized geometries with varying stenosis diameters and locations, employing multivariable linear regression (MLR), multi-layer perceptrons (MLPs), and convolutional neural networks (CNNs) as substitutes for computational fluid dynamics (CFD) to compute wall shear stress (WSS) during medical examinations. Although previous study (Su et al., 2020) described their input in Table 1 as a “fixed number of vessel’s centerline coordinates,” they effectively utilized point cloud data in a different coordinate system. Specifically, their input features included the centerline coordinates in the XY plane, along with the distance and angle relative to the centerline. Their study focused on a single coronary artery, varying morphological parameters such as vessel diameter, stenosis, and curvature.

To the best of our knowledge, researchers have proposed DL algorithm for predicting the TAWSS of bifurcations. However, no research has investigated DL-driven prediction on the cases varying in diameters and angulations using U-net structure. For this purpose, point cloud datasets are developed based on CFD simulations of ideal coronary bifurcations, including its hemodynamic parameters. Additionally, a CNN-based U-net is developed to generate the point cloud input for predicting TAWSS at each point. Finally, the effectiveness of the DL approach is assessed by evaluating the prediction errors in TAWSS.

Ordering point clouds is crucial, particularly in CNN architectures, as it helps extract local features. Principal Component Analysis (PCA) ordering is used which is one of the simplest methods for imposing order. This approach aligns the point cloud with its principal axes by computing eigenvectors of the covariance matrix. The points are then sorted based on their projections along the principal eigenvector, which corresponds to the direction of maximum variance. To assess its functionality, this ordering technique is compared with x, y, and z ordering points. As it is concluded from the Table 5, the MRE and NAME for ordering in canonical order is much better, 11.3 and 2.5, respectively.

As shown in Figure 3, one of the steps in the process involves reshaping 4,096 points into a 64 × 64 format. Specifically, the points are grouped into 64 clusters, each containing 64 points. After applying ordering to the point clouds, the grouping is schematically illustrated in Supplementary Figure S3.

In this study, our primary focus is on examining the hemodynamic parameter known as TAWSS, a metric previously investigated in studies related to coronary arteries (Eshtehardi et al., 2012; Su et al., 2020). Figure 4 compares the TAWSS values obtained from the CFD and DL models across two perspectives. Due to the curved geometry of the LAD and LCx in the idealized model, the TAWSS distribution is asymmetrical between the front and back views. Notably, the TAWSS values in the front view, as shown in Figures 4A, C, are almost 25% higher than those in the back views depicted in Figures 4B, D. In Figure 4A, left and right branches correspond to the LAD and LCx arteries, respectively. Although the values demonstrate good predictive capabilities in the LM and LAD vessels, the exact TAWSS values distribution from the CFD simulation (Figures 4A, B) has higher values in the bifurcation and LCx artery in this case.

Figure 5 depicts the comparison of TAWSS distribution between CFD and DL models along three lines on the wall within a bifurcation, representing specific paths of interest on the walls of LM, LAD, and LCx arteries. Each line consists of ten points, evenly spaced in the same direction as indicated by the arrow, to report the TAWSS value. This analysis evaluates the DL model’s ability to approximate the TAWSS distribution compared to the CFD approach from simulation. Generally, trends agree quite well in the models, however, in some points the absolute error is higher compared to other regions. For instance, in Figure 5C, it can be observed that in TAWSS graph along the LAD, in one point the difference reaches 20%.

Notably, the TAWSS value line graph for the LM artery in Figure 5B reduces from more than 15 [Pa] to 5 [Pa] at the outlet, with the DL model accurately predicting this trend. Moreover, TAWSS values along the LAD artery in Figure 5C remain relatively constant, fluctuating between 10 [Pa] and less than 12 [Pa] from the CFD and DL models, respectively. Additionally, there was a slight increase in TAWSS values along the line on the wall of LCx artery in the CFD model shown in Figure 5D, from 8 to 10 [Pa], with a less than 0.5 [Pa] difference in the DL model.

Figure 6 presents the results of TAWSS distribution obtained from CFD simulations (depicted in Figures 6A–C and the DL predictions (illustrated in Figures 6D–F, showing three different models with varying morphological parameters and associated errors. In all models, the left and right branches correspond to the LCx and LAD arteries, respectively. In branch region, the DL model tends to underestimate TAWSS values in LCx in Figure 6F). Nevertheless, it is evident that it is capable of capturing the overall pattern and values of TAWSS distribution.

Figure 7 presents a line graph comparing the TAWSS distribution in a region of interest on the LCx for varying α angles in two models. The TAWSS values are averaged across the region’s points, and the comparison is made between CFD and DL model outputs. The results show that the DL model accurately captures the overall increasing trend of TAWSS as the angle widens. Beyond 110°, TAWSS values plateau, although the DL model introduces minor variations. At smaller angles (below 70°), the differences between the models are minimal, remaining under 1.6 Pa, with the DL model closely matching the rising trend. However, at 75 and 105°, the model shows larger deviations, with discrepancies of 1.5 Pa and 1.1 Pa, respectively. While the DL model generally performs well in predicting TAWSS, fluctuating patterns in DL model suggest that further refinement is needed to improve its accuracy.

Su et al. (2020) achieved real-time 2D WSS prediction by creating numerous idealized blood vessel models derived from a limited clinical dataset and utilizing a CNN. It is shown in this study that CNN works better than MLP and MLR. However, their study does not include more complex geometries like bifurcations. Moreover, in their input they utilized 2D information of geometry which is much more straightforward than using a point cloud as an input. The Point Net structure, on the other hand, provides a significant advantage for geometries before and after treatment. Li et al. (2021a) and Wang et al. (2023b) applied the same structure for idealized geometry with cerebral aneurysm and patient-specific carotid artery geometries with stenosis, respectively. Despite its benefit that is robust to the number of input models, Point Net faces a prominent challenge in terms of variation in geometry features. As in the previous investigation, the Point Net model was trained separately for carotid arteries with and without stenosis (Wang S. et al., 2023).

However, in the proposed model not only the model is trained to predict data varying in the most crucial parameters like diameters and angels of bifurcation, but also it utilizes a point cloud as the input. The error function results for TAWSS at each mesh point, evaluated across the testing set models using the MRE Equation 5 and NAME Equation 6, were 11.39 ± 5.5 and 2.5 ± 0.56, respectively. It is important to note that the model was not previously trained on the testing set, which comprised 360 bifurcation models.

Due to the high dimensions of input data, CNN is a great tool for encoding the point cloud since it reduces trainable parameters rather than using MLP. In CNNs, each neuron in a layer is linked to a small region of the previous layer, known as the receptive field. This sparse connectivity minimizes redundancy, allowing the network to focus on local patterns. In the case study of Gharleghi et al. (Samarasinghe, 2020), they used CNN structure to predict the TAWSS in bifurcations, however, they used unfolding techniques to make 3D point clouds to 2D data. In case of bifurcation there is a complex relation between points in the center of geometry. Unfolding point clouds leads not to consider the relation between points where CNNs solely rely on local data. As a result, this structure cannot predict wide ranges of characteristics.

The proposed DL model for predicting TAWSS in coronary artery bifurcations offers substantial advantages for medical applications, particularly in real-time diagnostics and treatment planning. Traditional CFD simulations, though accurate, are computationally intensive and impractical for fast-response scenarios in clinical settings. Our DL-based approach, which predicts TAWSS with a normalized mean absolute error of 2.53% in under one second, overcomes this challenge by drastically enhancing computational efficiency without compromising accuracy. This advancement facilitates rapid, patient-specific assessments of hemodynamic factors linked to atherosclerosis and stenosis, enabling timely decision-making for both surgical and non-invasive interventions. Additionally, the model’s versatility across various coronary geometries further increases its clinical applicability, presenting a promising tool for personalized cardiovascular care.

A limitation we faced was the lack of patient-specific bifurcation models. To address this, we generated artificial bifurcation datasets for DL model, incorporating morphological parameters as substitutes for real-world data. Consequently, the optimal network parameters derived from the training samples deviated from actual clinical scenarios. Furthermore, during the CFD simulation process, we applied generic boundary conditions instead of tailored, patient-specific ones. Nevertheless, this approach is consistent with previous simulation studies (Davies et al., 2006; Jahromi et al., 2019). Another limitation was the use of a fixed number of point cloud as a input of the model, which could vary depending on the complexity of DL networks. Additionally, since our DL model’s input consisted of a uniform and fixed number of point cloud points across the geometries, some transitions in TAWSS values may have been overlooked in certain regions. Despite these constraints, we present a DL method that takes into account various restrictions, including dynamic flow boundary conditions. The DL model achieved an MRE value of 11.39 on the test dataset, successfully capturing the overall trend. However, a point-by-point analysis along the three vessels in Figure 5 reveals that the absolute relative error at one point reaches 20%. This result shows a limitation of the DL model, which could be addressed in future studies. Other parameters, such as velocity and pressure, are vector quantities, requiring predictions in three directions at each point. Additionally, velocity and pressure are defined for the fluid domain, making their complexity different from TAWSS, which is only defined on the wall. These aspects can be explored in future studies.

As a proof of concept, this study demonstrates the feasibility of using a DL-based approach for fast and accurate TAWSS prediction. The model achieved a NAME of only 2.5% and generate predictions in less than one second, underscoring its potential for rapid clinical applications. The efficiency gains, along with the strong correlation to CFD results, represent valuable contributions to the field.

In this study, we introduced a simulation-based framework for the first time to achieve Unet-based prediction of TAWSS in coronary arteries bifurcation with varied diameters and angels. Through the generation of high-quality point cloud datasets and the utilization of U-net, we demonstrated that the DL-based approach yields highly accurate predictions closely aligning with CFD simulations. Importantly, this achievement is accompanied by a notable reduction in computational costs. This emphasizes the potential and viability of the DL-based strategy for rapid and precise forecasting of TAWSS in coronary arteries bifurcation.