Figure 6 shows the cumulative energy rate as a function of the number of modes derived from the application of appropriate orthogonal decomposition (POD) methods at the ostium of the neck of an aneurysm. The plot shows four different variables: the directions of x, y, z, and the velocity components of pressure. From the graph, it is clear that a few modes can capture most of the dynamic energy of a system. For all velocity components (VEL_X, VEL_Y, VEL_Z), more than 90% of the total kinetic energy is captured in less than 10 modes. In particular, the X and Z velocity components converge faster than the Y components, indicating a more dominant coherent structure in their orientation in the neck region. In contrast, the pressure variables spike even further, reaching almost complete energy capture (>99%) in just a few modes, highlighting the relatively low dimension of complexity in this area.
The number of modes required for accumulated energy of velocity and pressure coefficients in the plane at the ostium.
Figure 7 shows the hemodynamic volume assessed in the aneurysm wall, in particular the percentage of accumulated energy and the number of pod modes for pressure, wall shear stress (WSS), and vibration shear index (OSI). This plot reveals that pressure and WSS achieve almost two energy captures in just a few modes. Both are to reach 99% of the accumulated energy in about the fifth mode. This suggests that their spatial distributions are highly structured and can be effectively reconstructed with minimal modal content, making them suitable for reduced modeling. In contrast, OSI slows down the energy storage curve. Starting at nearly 70%, more than 20 modes of more than 99% of the total energy are required, reflecting the higher complexity and spatial variability of the overall aneurysm wall. This is consistent with the sensitivity of OSI to changes in direction of wall shear on cardiac circulation, introducing a more dynamic mode into its representation.
The mode number required for the accumulated energy of OSI, WSS, and pressure coefficients on the SAC surface.
These results highlight that PODs can significantly reduce the computational load in capturing wall-related hemodynamic parameters, while retaining important information. In particular, high fidelity pressure and reconstruction of WSS fields can be performed using only a handful of modes. This is beneficial in accelerating the simulation workflow and enabling real-time or near-realistic predictions of clinical applications. Although a medium number of modes are required for more complex variables like OSI, the POD approach provides a manageable and efficient alternative to full resolution CFD simulations when assessing aneurysm rupture risk based on wall dynamics.
The results suggest that POD provides efficient dimensional reduction techniques and allows for a compact representation of flow dynamics in the aneurysm neck. This is especially useful when you can save only the most energetic modes, while still maintaining critical flow features and significantly reduce computational costs. Combined with machine learning or real-time simulation needs for patient-specific CFD workflows, it creates an attractive way. Based on the data achieved, the selected number of modes for the selected velocity components (VX, VY, VZ), pressure, WSS, and OSI are 8, 3, 5, and 9, respectively, to maintain 95% of the total energy of the model.
Figures 8a and B show reconstruction errors for various hemodynamic parameters. This was evaluated over time in two different regions of interest (the ostium in the neck (Fig. 8a) and the wall of the aneurysm (Fig. 8b). The vertical axis shows the difference (%) of L2 NORM on a logarithmic scale, and the relative reconstruction error between the original CFD data and the data reconstructed using a limited number of appropriate orthogonal decomposition (POD) modes. The horizontal axis represents the number of temporal snapshots used in dynamic simulations.
Data reconstruction error in a) Ostium plane b) On the SAC surface.
In Figure 8a, corresponding to the ostium in the neck, the pressure reconstruction error (black line) is the lowest of all quantities, consistently below 0.1%, reflecting excellent accuracy with minimal mode inclusion. Velocity components (VEL_X, VEL_Y, and VEL_Z) range from about 0.5% to 3%, with Vel_y (green) showing the highest variation. These small but non-negative variations suggest that the neck velocity field contains more dynamic complexity than pressure, but is still well captured by reduced models.
In contrast, Figure 8b, which represents the wall of an aneurysm, reveals more variable behavior. Again, pressure is the most accurately reconstructed variable, with reconstruction errors remaining below 0.1% in almost every snapshot. We also confirm that wall shear stress (WSS) is effectively represented in a relatively small pod mode, which usually maintains a low error of less than 1% and the dominant features of shear stress distribution are relatively small. However, the vibration shear index (OSI) shows a rather high reconstruction error that gradually drops above 100%, particularly during early snapshots.
Together, these numbers highlight that while pod-based low order modeling can robustly reconstruct pressure and WSS fields in both regions with high fidelity, OSI can derive greater modal resolutions based essentially on higher temporal and spatial complexity. This reinforces the idea that targeting variables like OSIs may be required for accurate real-time or machine learning-assisted predictions, while more stable quantities like pressure can be confidently approximated using compact, low-dimensional models.
Figure 9 consists of four subplots showing L2 norm reconstruction error (%) over time for different hemodynamic variables in the aneurysm neck ostium, using a model trained with appropriate orthogonal decomposition (POD) coupled with machine learning predictors. Each graph is useful for assessing model performance across the training (blue lines with diamond markers) and testing (green lines with circle markers) datasets, and for evaluating the generalization capabilities of the model.
Comparison of L2 norms of a)VX b)vy c)VZ and d) Pressure values for test and training datasets in the Ostium region of the neck.
Figure 9 confirms that the POD-ML approach can accurately reconstruct the flow dynamics of the aneurysm neck, especially due to pressure and specific velocity components. However, the growth of test errors in VEL_Y in particular emphasizes the importance of selecting the appropriate training interval and strengthening the model structure, enhancing the model structure (e.g., using recurrent neural networks or physics-based models) for more dynamic variables. These findings support the possibility of using reduced models for real-time predictions, and highlight the need for variable-specific modeling strategies to maintain reliability across the complete cardiac circulation.
Figure 10 shows a snapshot of three important hemodynamic variables assessed in the aneurysm wall over time: three important hemodynamic indexes (OSI), wall shear stress (WSS), and pressure. Each plot distinguishes training data (blue lines containing diamond markers) and test data (green lines with circle markers) and allows for the assessment of the generalization capabilities of a suitable orthogonal decomposition (POD) model with enhanced machine learning.
Comparison of L2 norms of a)OSI b)WSS and c) Pressure values for test and training datasets passing through the aneurysm wall.
Figure 10 collectively shows that pressure and WSS are suitable for pod-based low-order modeling and exhibit high accuracy in both the training and test regions. Figure 10b also shows the effects of various test train segmentations. As expected, reducing the test portion (corresponding to an increase in train data), lowering errors in the proposed method. In contrast, OSIs need to pay more attention, and more training data or enhanced modeling techniques (e.g., networks based on hybrid physics) must ensure accurate predictions across the time domain. These results have important implications for the simulation acceleration and real-time assessment of the wall state of aneurysms with reliable aneurysms, but OSI must be treated with a higher modeling refinement.
Comparison of a)VX b)vy c)VZ and d) Predict the contours of the reconstruction (left side) and predicted (right side) near the ostium section of park systolicity.
A comparison of the contours of reconstruction and prediction in sections near the ostium region is shown in Figure 11. The comparison of velocity and pressure in this diagram shows the efficiency of the technique of application in this diagram. Comparison of predicted velocity contours shows that although there is a large velocity gradient in the results, changes in velocity are predicted efficiently by the hybrid method. To assess the methodology, comparisons of OSI, WSS, and pressure on the SAC surface of the selected model were also made, and Figure 12 shows these contours. These hemodynamic factors are rationally predicted by our methodology.
Comparison of a)OSI b)WSS and c) Reconstructed pressure (left side) and predicted (right side) contours (right side) beyond the SAC surface area (right side) (it should be noted that WSS and pressure report data are obtained during peak systolic phase and OSI is calculated at the early diastolic phase).
Our model achieves high accuracy in the training set, but in fact there is an increase of 1-2 orders of magnitude increase in the test set. This is partially expected due to the complex nature of the OSI fields and the limited amount of training data available. However, for pressure and WSS, the test set error remains within acceptable range based on previous works17,25.
