Data-Driven Pulsatile Blood Flow Physics with Dynamic Mode Decomposition
Abstract
:1. Introduction
2. Methods
2.1. Computational Fluid Dynamics (CFD)
2.1.1. Cerebral Aneurysm
2.1.2. Coronary Artery Stenosis
2.2. Dynamic Mode Decomposition with Control (DMDc)
- Given data, construct snapshot matrices , , and . Then construct the augmented matrix based on and .
- Compute the SVD of the augmented matrix to get with truncation value q. Subsequently, based on the SVD results and the truncation value q, divide the complex conjugate transpose of into two distinct components .
- Compute the SVD of and obtain the truncated SVD decomposition with truncation value r ().
- Compute a reduced-order approximation of the linear operators and using:
- Compute the spectral decomposition of as to find its eigenvectors () and eigenvalues ().
- Extract the dynamic modes of the operator
2.3. Dynamic Mode Decomposition in Cardiovascular Flows
- Multistage dynamic mode decomposition with control (mDMDc)To model the effect of pulsatile flow ejected by the heart, the input matrix could be constructed as where is the mean velocity at the inlet and is computed from Q, the volumetric flow rate at the inlet boundary. Therefore, the mean velocity at the inlet is assumed to be the input controller. To account for different flow topologies in different parts of the cardiac cycle, we use multiple time windows and apply DMDc to each window separately. The different stages of the cardiac cycle are shown in Figure 2 where each stage is colored differently. The coronary artery stenosis flow rate is divided into three parts (low flow rate, acceleration, and deceleration). Since the blood flow in the aneurysm model is more complex and chaotic, we used six stages to capture distinct flow topologies.
- Augmented mDMDcTo model coherent structures (from velocity) and near-wall coherent structures (from WSS) separately, we consider an augmented matrix of velocity and WSS to construct the state space matrix as:Subsequently, the augmented multistage dynamic mode decomposition with control (mDMDc) leads to the following equation:
2.4. Performance Evaluation
3. Results
3.1. DMD and Womersley’s Analytical Solution
3.2. Data Reconstruction Accuracy
3.3. Flow Physics Using mDMDc
3.3.1. Flow Physics Using mDMDc Modes in Coronary Artery Stenosis
3.3.2. Flow Physics Using mDMDc Modes in Cerebral Aneurysm
4. Discussion
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Accuracy | 90% | 95% | 98% | 99% | 99.5% | (cm/s) | |
---|---|---|---|---|---|---|---|
Stage Number | |||||||
1st stage | 8 | 13 | 14 | 23 | 30 | 4.5 | |
2nd stage | 4 | 4 | 5 | 7 | 10 | 7.6 | |
3rd stage | 2 | 2 | 3 | 3 | 8 | 7.8 |
Accuracy | 90% | 95% | 98% | 99% | 99.5% | (dynes/cm2) | |
---|---|---|---|---|---|---|---|
Stage Number | |||||||
1st stage | 10 | 13 | 23 | 32 | 41 | 1.5 | |
2nd stage | 4 | 5 | 6 | 8 | 18 | 3.0 | |
3rd stage | 2 | 2 | 3 | 4 | 10 | 3.1 |
Accuracy | 90% | 95% | 98% | 99% | 99.5% | (cm/s) | |
---|---|---|---|---|---|---|---|
Stage Number | |||||||
1st stage | 3 | 3 | 4 | 5 | 12 | 7.1 | |
2nd stage | 2 | 2 | 3 | 5 | 6 | 17.8 | |
3rd stage | 5 | 5 | 7 | 13 | 22 | 22.1 | |
4th stage | 3 | 3 | 4 | 9 | 20 | 19.2 | |
5th stage | 3 | 6 | 6 | 10 | 18 | 14.1 | |
6th stage | 5 | 5 | 13 | 31 | 48 | 8.4 |
Accuracy | 90% | 95% | 98% | 99% | 99.5% | (dynes/cm2) | |
---|---|---|---|---|---|---|---|
Stage Number | |||||||
1st stage | 3 | 3 | 4 | 5 | 19 | 23.6 | |
2nd stage | 2 | 2 | 3 | 5 | 6 | 78.5 | |
3rd stage | 5 | 10 | 10 | 13 | 25 | 105.6 | |
4th stage | 7 | 7 | 7 | 9 | 20 | 86.9 | |
5th stage | 3 | 3 | 4 | 9 | 17 | 56.6 | |
6th stage | 5 | 13 | 13 | 31 | 53 | 3.6 |
Accuracy | 90% | 95% | 98% | 99% | 99.5% | (cm/s) |
---|---|---|---|---|---|---|
Coronary artery stenosis | 27 | 41 | 53 | 77 | 88 | 6.7 |
Cerebral aneurysm | 96 | 110 | 155 | 165 | 173 | 12.1 |
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Habibi, M.; Dawson, S.T.M.; Arzani, A. Data-Driven Pulsatile Blood Flow Physics with Dynamic Mode Decomposition. Fluids 2020, 5, 111. https://doi.org/10.3390/fluids5030111
Habibi M, Dawson STM, Arzani A. Data-Driven Pulsatile Blood Flow Physics with Dynamic Mode Decomposition. Fluids. 2020; 5(3):111. https://doi.org/10.3390/fluids5030111
Chicago/Turabian StyleHabibi, Milad, Scott T. M. Dawson, and Amirhossein Arzani. 2020. "Data-Driven Pulsatile Blood Flow Physics with Dynamic Mode Decomposition" Fluids 5, no. 3: 111. https://doi.org/10.3390/fluids5030111