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Edited by: Rajat Mittal, Johns Hopkins University, United States

Reviewed by: Wenjun Kou, Northwestern University, United States; Daniel Goldman, Western University, Canada; Gustavo Boroni, Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina

^{†}These authors have contributed equally to this work and share first authorship

This article was submitted to Computational Physiology and Medicine, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The interventional treatment of cerebral aneurysm requires hemodynamics to provide proper guidance. Computational fluid dynamics (CFD) is gradually used in calculating cerebral aneurysm hemodynamics before and after flow-diverting (FD) stent placement. However, the complex operation (such as the construction and placement simulation of fully resolved or porous-medium FD stent) and high computational cost of CFD hinder its application. To solve these problems, we applied aneurysm hemodynamics point cloud data sets and a deep learning network with double input and sampling channels. The flexible point cloud format can represent the geometry and flow distribution of different aneurysms before and after FD stent (represented by porous medium layer) placement with high resolution. The proposed network can directly analyze the relationship between aneurysm geometry and internal hemodynamics, to further realize the flow field prediction and avoid the complex operation of CFD. Statistical analysis shows that the prediction results of hemodynamics by our deep learning method are consistent with the CFD method (error function <13%), but the calculation time is significantly reduced 1,800 times. This study develops a novel deep learning method that can accurately predict the hemodynamics of different cerebral aneurysms before and after FD stent placement with low computational cost and simple operation processes.

Currently, strokes, including cerebral aneurysms, have the highest mortality rate of diseases worldwide (

Computational fluid dynamics (CFD) is gradually used in the calculation of hemodynamics. Based on the given boundary conditions and geometric information of the model, CFD can solve the conservation equations of mass, momentum and energy on the discrete meshes by the Navier–Stokes equation, and then obtain the numerical solutions of the flow field hemodynamics. However, when calculating the hemodynamics of cerebral aneurysm with FD stent, CFD often needs complex operation processes and high computational costs. This is due to the construction and placement simulation of a fully resolved FD stent, which usually requires professional operation skills and long-time iterative calculation (

Machine learning or deep learning technique has been used for fluid dynamics field such as modeling of turbulent flow or further direct estimation of flow field. With the support of high-performance GPU computing clusters and network structures, deep learning can extract the underlying features through neural network to build abstract high-level features or attribute features, and then achieve faster and more accurate pattern classifications or regression tasks than traditional methods (

In this study, we evolved the hemodynamics prediction network for the cerebral aneurysms flow including FD stent layers. We established the aneurysm hemodynamics point cloud data sets based on CFD simulation of ideal side-wall aneurysms with or without porous-medium models. Corresponding to the data set, a deep learning network with double input and sampling channels was proposed. The velocity and pressure prediction errors were calculated to evaluate deep learning performance. Compared with the previous deep learning method, our deep learning method can flexibly characterize the complex geometry including fluid domain and FD domain in the same scheme and realize the flow field prediction without identifying the point cloud belonging to FD domain. Compared with the traditional CFD approach to calculate the aneurysm hemodynamics with FD stent, our deep learning method could avoid the stent construction and placement simulation, which greatly simplified the operation process and reduces the computational cost.

The data sets used in this study were from the CFD simulation hemodynamics of the side-wall aneurysms before and after FD stent placement. Therefore, the creation of data sets included three steps: the generation of cerebral aneurysm models, the flow field calculation of cerebral aneurysm model before and after FD stent placement, the point clouds extraction and data sets establishment.

We divided the models into two types according whether the parent artery was straight or curved, as shown in

Definition of morphological parameters and generation of cerebral aneurysm models.

The range of morphological parameters.

^{–}^{1}) |
|||||

Range | 8.0–12.0 | 6.0–8.0 | 8.0–12.0 | 0.01–0.033 | 0–180° |

The flow field hemodynamic inside the aneurysm model was calculated by the CFD method. Under the assumptions of incompressible Newtonian laminar flow, constant density and no slip wall, the continuity equation and Navier–Stokes equation were solved to calculate the flow field velocity and pressure:

Where _{x}, _{y}, and _{z} were the velocity components in the flow field in x, y, and z directions, respectively. P was the flow field pressure, ρ was the fluid density, and μ was the fluid viscosity. Regarding the material properties of vessels and blood, ρ was 1,050 kg/m^{3} and μ was 0.0035 Pa⋅s.

In this study, the simulation was set as a steady simulation, which meant that boundary conditions were constants independent of time. The purpose of this study was to use deep learning to analyze and reproduce the relationship between model geometry and hemodynamics. Therefore, we set the boundary conditions within a reasonable range. The boundary conditions of all models were uniformly set as 0.004375 kg/s at the inlet and zero pressure at the outlet.

As mentioned above, we used porous media method to represent FD stent. The shape of the porous media layer was based on the shape of the surface of the blood vessel wall at the intersection plane between the sphere and the parent artery. The thickness of the porous media layer was 150 μm. The resistance S of the porous media layer could be expressed as:

Where the specific values of permeability K and loss coefficient C were 0.001489 mm^{2} and 7,665 m^{–1}, which was obtained according to FD stent properties (

All models were established using Solidworks (France) commercial software. After the 3D models were pre-processed, ANSYS-Meshing (United States) commercial software was used for making mesh to generate the computational models. In the final step, we used ANSYS-CFX for simulation and got the velocity and pressure distribution results.

Using ANSYS or other simulation software, the hemodynamic results from CFD could be directly transformed into a point cloud format. The point cloud was extracted from the connection points (usually called nodes) of the CFD meshes, so it inherited the CFD meshes to resolve the geometric structure of the model. The spatial distribution of the point cloud varied with the geometry of different models. Compared with the fixed format samples that were used in the previous deep learning methods (

Two kinds of point clouds were extracted—namely, the model point cloud{_{i}|_{1}} and query point cloud{_{i}|_{2}}. _{1} and _{2} were the total number of model points and query points in one model, respectively. The model point cloud was extracted from the outermost mesh nodes, which contained only the overall geometric information (the set of coordinates in x, y, and z directions of each point) of the cerebral aneurysm. The query point cloud was extracted from the mesh nodes inside the model. In addition to the spatial distribution information, the query point cloud also contained the velocity and pressure value corresponding to each point.

We extracted two kinds of point clouds of all models before and after FD stent placement and then constructed four aneurysm hemodynamic data sets: preoperative, postoperative, velocity, and pressure fields data sets. We randomly divided each data set into a training set (a total of 450, including 90 type 1 and 360 type 2 models) and a test set (a total of 50, including 10 type 1 and 40 type 2 models) according to the proportion of 9:1. During the pre-experiment, we calculated the prediction error of multiple randomly divided training/test set combinations (the proportion is 9:1) on the network. There was no significant difference in the errors of different data sets, which could prove the robustness of the scheme. The four data sets needed to be used separately as inputs to train the network. Velocity or pressure predictions also needed to be carried out separately using the corresponding trained network.

Based on the properties of created data sets, we adopted an optimized network structure for flow prediction on flexible structure of point cloud (

The proposed network structure. N (in millions) represents the total number of points contained in the query point cloud and the model point cloud in one model.

The training process and operation principle of our network could be described as follows:

Where _{1}(_{H1},_{H2},…,_{HN2}) represented the predicted output of hemodynamics corresponding to each point of the input query point cloud. In the training process, query point cloud {_{i}|_{2}} (x, y, z coordinates and corresponding velocity or pressure value) and model point cloud {_{i}|_{1}} (x, y, z coordinates) were used as network input. _{1} and _{2} represented the feature extraction methods represented by module 2 and module 1, respectively. _{1} extracted the local coordinates and corresponding hemodynamic features of each point in the input query point cloud, which were used as teaching signals in the training process. _{2} extracted the overall geometric features represented by the input model point cloud. G was the symmetric function (MaxPooling). The addition of _{1} and _{2} represented the feature stitching operation, that was, the two features were stitched together to realize the further prediction function of the output layer. γ represented high-dimensional abstract feature extraction. Furthermore:

The proposed deep learning network consisted of four parts: input layers, feature extraction layers, feature stitching layers, and output layers.

Input layers: this part included two input layers, which were used to import the model point cloud and query point cloud of aneurysm, respectively.

Feature extraction layers: two feature extraction modules were directly connected with two input layers. Module 1 extracted the geometric features of aneurysm as global features. Module 2 extracted the spatial coordinates of the internal query point cloud and the corresponding flow velocity or pressure field as local features. In order to enhance the relationship between global features and local features, fully connected layer 1 (FC1) and fully connected layer 2 (FC2) shared the weights, which meant that they shared the same extraction method in the primary extraction stage. The symmetric function Maxpooling layer after module 1 could solve the disorder of input point cloud (

Feature stitching layers: In this part, vectors representing global and local features could be stitched together to form an N × 640 (512+128) dimensional feature. With the global feature as the constraint and the local feature as the teachers’ signal, the spatial relationship was effectively introduced to help the network attain correspondence between the model geometry and the flow velocity or pressure distribution point by point.

Output layers: this part outputted the flow velocity field corresponding to the internal query point cloud.

The difference between our deep learning method and the original PointNet was that there were two types of input point clouds and two corresponding input layers of the network. The original PointNet had only one type of input point cloud and a single input layer. Our design could make the network extract the geometric structure and flow field features of the training data better, which was proved by an ablation experiment (comparing the prediction error between our network and the original PointNet) and a control experiment (with or without “shared weight”) in our previous study (

For other details, we used the mean absolute error as the loss function. Since the number of points in a single model reached the level of 0.3 million, the parameter batch size was set to 1, which meant that the input for one training iteration was all model points and query points in a single model. The optimizer was Adam with learning rate = 0.001, ε = 0.001, ρ1 = 0.9, ρ2 = 0.999, and δ = 1E–8 (

The usage details of data sets during network training and testing.

Training | Training set | Model point cloud (x, y, z coordinates) + query point cloud (x, y, z coordinates and hemodynamic values calculated by CFD) | Deep learning hemodynamic prediction values corresponding to input query point cloud | 450 | About 0.1 (Type 1 model) –0.3 (Type 2 model) million |

Testing | Test set | Model point cloud (x, y, z coordinates) + query point cloud (only x, y, z coordinates) | Deep learning hemodynamic prediction values corresponding to input query point cloud | 50 | About 0.1 (Type 1 model) –0.3 (Type 2 model) million |

In order to evaluate the performance of our deep learning method, referring to the previous research (

Where _{i} and

Due to the relatively simple geometry and internal flow field of the type 1 models, we took the type 2 models as examples to show the mesh independence test result, as shown in

The mesh independence test.

To visually show the prediction results of deep learning, we randomly selected the models in the test sets and set the cross section, as shown in

Sketch of measurement planes. For Type 2 models, use a sample with an LA of 30°. The results shown in

Comparison of velocity fields from CFD and deep learning (DL) methods.

Comparison of pressure fields from CFD and deep learning (DL) methods.

The velocity field and pressure field in the selected section are shown in

Deep learning predicted results and CFD results were in good agreement. In terms of flow field properties, our deep learning method could not only predict the laminar properties of the blood flow in the parent artery, but also predict the generation of complex vortexes inside the aneurysm. In terms of the stage before and after FD stent placement, deep learning could predict the distribution of flow velocity and pressure field before operation, and reflect its change after operation.

ERRs statistics results.

Preoperative | Type 1 | Velocity | Whole model | 3.46 ± 1.21 | 7.52 ± 1.33 |

Aneurysm | 3.71 ± 2.31 | 7.62 ± 2.01 | |||

Pressure | Whole model | 2.21 ± 1.35 | 6.35 ± 2.13 | ||

Aneurysm | 2.97 ± 2.31 | 7.01 ± 1.97 | |||

Type 2 | Velocity | Whole model | 4.31 ± 1.95 | 9.81 ± 1.59 | |

Aneurysm | 6.37 ± 2.81 | 12.97 ± 3.26 | |||

Pressure | Whole model | 3.54 ± 2.03 | 6.82 ± 2.51 | ||

Aneurysm | 5.01 ± 2.75 | 9.01 ± 3.12 | |||

Postoperative | Type 1 | Velocity | Whole model | 3.71 ± 2.01 | 7.84 ± 1.68 |

Aneurysm | 3.75 ± 1.38 | 8.01 ± 1.52 | |||

Pressure | Whole model | 2.17 ± 1.41 | 7.01 ± 1.69 | ||

Aneurysm | 2.21 ± 1.54 | 7.15 ± 1.34 | |||

Type 2 | Velocity | Whole model | 4.27 ± 1.87 | 9.92 ± 1.98 | |

Aneurysm | 5.01 ± 2.17 | 11.21 ± 2.10 | |||

Pressure | Whole model | 3.71 ± 2.21 | 6.94 ± 1.26 | ||

Aneurysm | 4.25 ± 1.98 | 7.31 ± 2.15 |

As for computational cost, hemodynamics of one model could be obtained within 1 s through deep learning. For the CFD method, including the construction and placement simulation operation of porous-medium model of the FD stent and calculation of hemodynamics, it took about 30 min on an Intel Xeon Gold 6148 2.4 GHz × 2 CPU server. Deep learning could reduce the calculation time 1,800 times.

In this study, we proposed a deep learning method to predict the hemodynamics of cerebral aneurysms before and after the FD stent placement. Compared with the previous deep learning method, our deep learning method could flexibly characterize the complex geometry including fluid domain and FD domain in the same scheme. And it realized the flow field prediction without identifying the point cloud belonging to FD domain. The results of error analysis showed that our deep learning method could achieve hemodynamic prediction with high accuracy (ERR < 13%) while reducing the calculation time by 1,800 times. It showed the practical value of deep learning in the task of calculating hemodynamics, which could greatly reduce the computational cost and simplify the operation process.

We conducted a comparative analysis of the previous studies on the prediction of flow field or hemodynamic parameters via machine or deep learning (

Flow or hemodynamics prediction accuracy reported by other machine or deep learning studies.

The proposed deep learning method | 3D cerebral aneurysm hemodynamics | 500 | Flexible point cloud | NMAE < 6.5%, MRE < 13% |

Itu’s machine-learning model | Fractional flow reserve (FFR) value | 12,000 | Geometric parameter | Error = 0.03% |

Lee’s CNNs | 2D unsteady flow field | 500,000 | Fixed meshes | 32.8% < Error < 1% |

Guo’s DCNNs | 2D/3D steady flow | 400,000 | Fixed pixels | MRE < 3% |

Liang’s DNNs | 3D thoracic aorta hemodynamics | 729 | Fixed meshes | NMAE < 6.5% |

The use of machine learning or deep learning for hemodynamic parameters (such as FFR) and flow field prediction was still limited.

The ERRs of type 2 models were higher than that of type 1. This might be due to the different complexity of the parent artery flow field. To fully develop the flow field, we constructed a long parent artery. The number of parent artery query points accounted for about 90% of the whole model, which meant that the ERRs of the whole model were mainly determined by the parent artery. In type 2 models, secondary flow due to curvature happened in the parent artery (

The ERRs of the aneurysm were higher than that of the whole model. The flow field inside the aneurysm was mainly determined by the inflow flux at the aneurysm orifice. As

This study had several limitations. First, there was a lack of data for real patients’ cerebral aneurysms. We built the deep learning aneurysm data sets by combining morphological parameters to replace the real model. This inevitably led to the deviation of the optimal parameter configuration of the network obtained from the training samples from the real situation. Secondly, we only selected the side-wall aneurysms on cavernous branches as the object, and lack studies on other types of cerebral aneurysms (such as fusiform aneurysm, etc.). In further work, we will collect different types of cerebral aneurysm models from real patients to verify the applicability of our deep learning method. Thirdly, in the CFD simulation process, we used general boundary conditions instead of personalized boundary conditions. In addition, we used one kind of porous media model to replace the real stent. This may also lead to a potential risk, because the influence of the porous media model and the real stent model on hemodynamics was different. However, this parameter setting strategy was widely used in lots of simulation studies (

In this study, we built flexible, high-resolution point cloud data sets and propose the corresponding network structure. On this basis, we realized the fast and accurate prediction of cerebral aneurysm hemodynamics before and after stent placement.

Compared with the traditional CFD method, our deep learning method can greatly simplify the operation process and reduce the computational cost. Compared with previous deep learning research, our deep learning method has more flexible data format and higher resolution, which means higher versatility and can be applied to the flow field prediction of other human parts or even other research fields. In terms of data resolution, computational efficiency, and universality, our deep learning method can meet the needs of most situations.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

GL, XS, and XW created and designed this study. GL, HW, and SL performed the simulation and analyzed the data. All authors discussed and co-authored the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

This work was supported by the National Natural Science Foundation of China (11772015, 11832003, and 11772016).