# Data-Driven Pulsatile Blood Flow Physics with Dynamic Mode Decomposition

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## Abstract

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## 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 $\mathbf{X}$, ${\mathbf{X}}^{\prime}$, and $\mathsf{\Psi}$. Then construct the augmented matrix $\mathsf{\Omega}$ based on $\mathbf{X}$ and $\mathsf{\Psi}$.
- Compute the SVD of the augmented matrix $\mathsf{\Omega}$ to get $\mathsf{\Omega}\approx \tilde{\mathbf{U}}\tilde{\mathsf{\Sigma}}{\tilde{\mathbf{V}}}^{*}$ with truncation value q. Subsequently, based on the SVD results and the truncation value q, divide the complex conjugate transpose of $\tilde{\mathbf{U}}$ into two distinct components ${\tilde{\mathbf{U}}}^{*}=[{\tilde{\mathbf{U}}}_{1}^{*},{\tilde{\mathbf{U}}}_{2}^{*}]$.
- Compute the SVD of ${\mathbf{X}}^{\prime}$ and obtain the truncated SVD decomposition ${\mathbf{X}}^{\prime}\approx \widehat{\mathbf{U}}\widehat{\mathsf{\Sigma}}{\widehat{\mathbf{V}}}^{*}$ with truncation value r ($\widehat{\mathsf{\Sigma}}\in {\mathbb{R}}^{r\times r}$).
- Compute a reduced-order approximation of the linear operators $\mathbf{A}$ and $\mathbf{B}$ using:$$[\widehat{\mathbf{A}},\widehat{\mathbf{B}}]=[{\widehat{\mathbf{U}}}^{*}{\mathbf{X}}^{\prime}\tilde{\mathbf{V}}{\tilde{\mathsf{\Sigma}}}^{-\mathbf{1}}{\tilde{\mathbf{U}}}_{1}^{*}\widehat{\mathbf{U}},{\widehat{\mathbf{U}}}^{*}{\mathbf{X}}^{\prime}\tilde{\mathbf{V}}{\tilde{\mathsf{\Sigma}}}^{-\mathbf{1}}{\tilde{\mathbf{U}}}_{2}^{*}]\phantom{\rule{0.277778em}{0ex}},$$
- Compute the spectral decomposition of $\tilde{\mathbf{A}}$ as $\tilde{\mathbf{A}}\mathbf{W}=\mathbf{W}\mathsf{\Lambda}$ to find its eigenvectors ($\mathbf{W}$) and eigenvalues ($\mathsf{\Lambda}$).
- Extract the dynamic modes of the operator $\mathbf{A}$$$\mathsf{\Phi}={\mathbf{X}}^{\prime}\tilde{\mathbf{V}}{\tilde{\mathsf{\Sigma}}}^{-\mathbf{1}}{\tilde{\mathbf{U}}}_{1}^{*}\widehat{\mathbf{U}}\mathbf{W}\phantom{\rule{0.277778em}{0ex}}.$$

#### 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 $\mathsf{\Psi}=\left[\begin{array}{cccc}{V}_{1}^{mean}& {V}_{2}^{mean}& \cdots & {V}_{N-1}^{mean}\end{array}\right]$ where ${V}^{mean}=Q/{A}_{inlet}$ 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 mDMDc**To 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:$$\mathbf{X}=\left[\begin{array}{cccc}{\mathbf{V}}_{1}& {\mathbf{V}}_{2}& \cdots & {\mathbf{V}}_{N-1}\\ \phantom{\rule{4pt}{0ex}}{\mathbf{WSS}}_{1}& {\mathbf{WSS}}_{2}& \cdots & {\mathbf{WSS}}_{N-1}\end{array}\right],{\mathbf{X}}^{\prime}=\left[\begin{array}{cccc}{\mathbf{V}}_{2}& {\mathbf{V}}_{3}& \cdots & {\mathbf{V}}_{N}\\ \phantom{\rule{4pt}{0ex}}{\mathbf{WSS}}_{2}& {\mathbf{WSS}}_{3}& \cdots & {\mathbf{WSS}}_{N}\end{array}\right]\phantom{\rule{0.277778em}{0ex}}.$$Subsequently, the augmented multistage dynamic mode decomposition with control (mDMDc) leads to the following equation:$${\mathbf{X}}^{\prime}=\mathbf{A}\mathbf{X}+\mathbf{B}\mathsf{\Psi}\phantom{\rule{0.277778em}{0ex}},$$$$\begin{array}{c}\hfill \left[\begin{array}{cccc}{\mathbf{V}}_{2}& {\mathbf{V}}_{3}& \cdots & {\mathbf{V}}_{N}\\ \phantom{\rule{4pt}{0ex}}{\mathbf{WSS}}_{2}& {\mathbf{WSS}}_{3}& \cdots & {\mathbf{WSS}}_{N}\end{array}\right]=\mathbf{A}\left[\begin{array}{cccc}{\mathbf{V}}_{1}& {\mathbf{V}}_{2}& \cdots & {\mathbf{V}}_{N-1}\\ \phantom{\rule{4pt}{0ex}}{\mathbf{WSS}}_{1}& {\mathbf{WSS}}_{2}& \cdots & {\mathbf{WSS}}_{N-1}\end{array}\right]\\ \hfill +\mathbf{B}\left[\begin{array}{cccc}{V}_{1}^{mean}& {V}_{2}^{mean}& \cdots & {V}_{N-1}^{mean}\end{array}\right]\phantom{\rule{0.277778em}{0ex}}.\end{array}$$

#### 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

- Taylor, C.A.; Figueroa, C.A. Patient-specific modeling of cardiovascular mechanics. Ann. Rev. Biomed. Eng.
**2009**, 11, 109–134. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Morris, P.D.; Narracott, A.; von Tengg-Kobligk, H.; Soto, D.A.S.; Hsiao, S.; Lungu, A.; Evans, P.; Bressloff, N.W.; Lawford, P.V.; Hose, D.R.; et al. Computational fluid dynamics modelling in cardiovascular medicine. Heart
**2016**, 102, 18–28. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Taylor, C.A.; Fonte, T.A.; Min, J.K. Computational fluid dynamics applied to cardiac computed tomography for noninvasive quantification of fractional flow reserve: Scientific basis. J. Am. Coll. Cardiol.
**2013**, 61, 2233–2241. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Khan, M.O.; Valen-Sendstad, K.; Steinman, D.A. Narrowing the expertise gap for predicting intracranial aneurysm hemodynamics: Impact of solver numerics versus mesh and time-step resolution. Am. J. Neuroradiol.
**2015**, 36, 1310–1316. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Arzani, A. Accounting for residence-time in blood rheology models: Do we really need non-Newtonian blood flow modeling in large arteries? J. R. Soc. Interface
**2018**, 15, 20180486. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Taira, K.; Brunton, S.L.; Dawson, S.T.M.; Rowley, C.W.; Colonius, T.; McKeon, B.J.; Schmidt, O.T.; Gordeyev, S.; Theofilis, V.; Ukeiley, L.S. Modal analysis of fluid flows: An overview. AIAA J.
**2017**, 102, 4013–4041. [Google Scholar] [CrossRef] [Green Version] - Brunton, S.L.; Noack, B.R.; Koumoutsakos, P. Machine learning for fluid mechanics. Ann. Rev. Fluid Mech.
**2020**, 52, 477–508. [Google Scholar] [CrossRef] [Green Version] - Bamieh, B.; Giarre, L. Identification of linear parameter varying models. Int. J. Robust Nonlinear Control
**2002**, 12, 841–853. [Google Scholar] [CrossRef] - Paoletti, S.; Juloski, A.; Ferrari-Trecate, G.; Vidal, R. Identification of hybrid systems: A tutorial. Eur. J. Control
**2007**, 13, 242–260. [Google Scholar] [CrossRef] [Green Version] - Kutz, J.N.; Brunton, S.L.; Brunton, B.W.; Proctor, J.L. Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems; SIAM: Philadelphia, PA, USA, 2016. [Google Scholar]
- Mirramezani, M.; Diamond, S.L.; Litt, H.I.; Shadden, S.C. Reduced order models for transstenotic pressure drop in the coronary arteries. J. Biomech. Eng.
**2019**, 141, 031005. [Google Scholar] [CrossRef] - Sankaran, S.; Lesage, D.; Tombropoulos, R.; Xiao, N.; Kim, H.J.; Spain, D.; Schaap, M.; Taylor, C.A. Physics driven reduced order model for real time blood flow simulations. arXiv
**2019**, arXiv:1911.01543. [Google Scholar] - Berkooz, G.; Holmes, P.; Lumley, J.L. The proper orthogonal decomposition in the analysis of turbulent flows. Ann. Rev. Fluid Mech.
**1993**, 25, 539–575. [Google Scholar] [CrossRef] - Holmes, P.; Lumley, J.L.; Berkooz, G.; Rowley, C.W. Turbulence, Coherent Structures, Dynamical Systems and Symmetry; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Glenn, A.L.; Bulusu, K.V.; Shu, F.; Plesniak, M.W. Secondary flow structures under stent-induced perturbations for cardiovascular flow in a curved artery model. Int. J. Heat Fluid Flow
**2012**, 35, 76–83. [Google Scholar] [CrossRef] - Chang, G.H.; Schirmer, C.M.; Modarres-Sadeghi, Y. A reduced-order model for wall shear stress in abdominal aortic aneurysms by proper orthogonal decomposition. J. Biomech.
**2017**, 54, 33–43. [Google Scholar] [CrossRef] - Grinberg, L.; Yakhot, A.; Karniadakis, G. Analyzing transient turbulence in a stenosed carotid artery by proper orthogonal decomposition. Ann. Biomed. Eng.
**2009**, 37, 2200–2217. [Google Scholar] [CrossRef] [PubMed] - Kefayati, S.; Poepping, T.L. Transitional flow analysis in the carotid artery bifurcation by proper orthogonal decomposition and particle image velocimetry. Med. Eng. Phys.
**2013**, 35, 898–909. [Google Scholar] [CrossRef] - Wold, S.; Esbensen, K.; Geladi, P. Principal component analysis. Chemom. Intell. Lab. Syst.
**1987**, 2, 37–52. [Google Scholar] [CrossRef] - Schmid, P.J. Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech.
**2010**, 656, 5–28. [Google Scholar] [CrossRef] [Green Version] - Noack, B.R. From snapshots to modal expansions–bridging low residuals and pure frequencies. J. Fluid Mech.
**2016**, 802, 1–4. [Google Scholar] [CrossRef] [Green Version] - Rowley, C.W.; Mezić, I.; Bagheri, S.; Schlatter, P.; Henningson, D.S. Spectral analysis of nonlinear flows. J. Fluid Mech.
**2009**, 641, 115–127. [Google Scholar] [CrossRef] [Green Version] - Mezić, I. Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn.
**2005**, 41, 309–325. [Google Scholar] [CrossRef] - Perko, L. Differential Equations and Dynamical Systems; Springer Science & Business Media: Berlin, Germany, 2013. [Google Scholar]
- Wynn, A.; Pearson, D.; Ganapathisubramani, B.; Goulart, P. Optimal mode decomposition for unsteady flows. J. Fluid Mech.
**2013**, 733, 473–503. [Google Scholar] [CrossRef] [Green Version] - Jovanović, M.; Schmid, P.; Nichols, J. Sparsity-promoting dynamic mode decomposition. Phys. Fluids
**2014**, 26, 024103. [Google Scholar] [CrossRef] - Proctor, J.; Brunton, S.; Kutz, J. Dynamic mode decomposition with control. SIAM J. Appl. Dyn. Syst.
**2016**, 15, 142–161. [Google Scholar] [CrossRef] [Green Version] - Kutz, J.N.; Fu, X.; Brunton, S. Multiresolution dynamic mode decomposition. SIAM J. Appl. Dyn. Syst.
**2016**, 15, 713–735. [Google Scholar] [CrossRef] [Green Version] - Dawson, S.T.M.; Hemati, M.S.; Williams, M.O.; Rowley, C. Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition. Exp. Fluids
**2016**, 57, 42. [Google Scholar] [CrossRef] [Green Version] - Annoni, J.; Seiler, P. A method to construct reduced-order parameter-varying models. Int. J. Robust Nonlinear Control
**2017**, 27, 582–597. [Google Scholar] [CrossRef] - Kou, J.; Zhang, W. Dynamic mode decomposition with exogenous input for data-driven modeling of unsteady flows. Phys. Fluids
**2019**, 31, 057106. [Google Scholar] - Lu, H.; Tartakovsky, D.M. Lagrangian Dynamic Mode Decomposition for Construction of Reduced-Order Models of Advection-Dominated Phenomena. J. Comput. Phys.
**2020**, 407, 109229. [Google Scholar] [CrossRef] [Green Version] - Schmid, P.J.; Li, L.; Juniper, M.P.; Pust, O. Applications of the dynamic mode decomposition. Theor. Comput. Fluid Dyn.
**2011**, 25, 249–259. [Google Scholar] [CrossRef] - Seena, A.; Sung, H.J. Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations. Int. J. Heat Fluid Flow
**2011**, 32, 1098–1110. [Google Scholar] [CrossRef] - Dawson, S.T.M.; Schiavone, N.; Rowley, C.; Williams, D. A data-driven modeling framework for predicting forces and pressures on a rapidly pitching airfoil. In Proceedings of the 45th AIAA Fluid Dynamics Conference, Dallas, TX, USA, 22–26 June 2015; p. 2767. [Google Scholar]
- Le Clainche, S.; Han, Z.H.; Ferrer, E. An alternative method to study cross-flow instabilities based on high order dynamic mode decomposition. Phys. Fluids
**2019**, 31, 094101. [Google Scholar] [CrossRef] - Alessandri, A.; Bagnerini, P.; Gaggero, M.; Lengani, D.; Simoni, D. Dynamic mode decomposition for the inspection of three-regime separated transitional boundary layers using a least squares method. Phys. Fluids
**2019**, 31, 044103. [Google Scholar] [CrossRef] - Pain, R.; Weiss, P.E.; Deck, S.; Robinet, J.C. Large scale dynamics of a high Reynolds number axisymmetric separating/reattaching flow. Phys. Fluids
**2019**, 31, 125119. [Google Scholar] [CrossRef] - Delorme, Y.; Kerlo, A.; Anupindi, K.; Rodefeld, M.; Frankel, S. Dynamic mode decomposition of Fontan hemodynamics in an idealized total cavopulmonary connection. Fluid Dyn. Res.
**2014**, 46, 041425. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Di Labbio, G.; Kadem, L. Reduced-order modeling of left ventricular flow subject to aortic valve regurgitation. Phys. Fluids
**2019**, 31, 031901. [Google Scholar] [CrossRef] [Green Version] - Arzani, A.; Shadden, S.C. Characterization of the transport topology in patient-specific abdominal aortic aneurysm models. Phys. Fluids
**2012**, 24, 081901. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Womersley, J.R. Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol.
**1955**, 127, 553–563. [Google Scholar] [CrossRef] - Mortensen, M.; Valen-Sendstad, K. Oasis: A high-level/high-performance open source Navier–Stokes solver. Comp. Phys. Commun.
**2015**, 188, 177–188. [Google Scholar] [CrossRef] [Green Version] - Valen-Sendstad, K.; Steinman, D.A. Mind the gap: Impact of computational fluid dynamics solution strategy on prediction of intracranial aneurysm hemodynamics and rupture status indicators. Am. J. Neuroradiol.
**2014**, 35, 536–543. [Google Scholar] [CrossRef] [Green Version] - Updegrove, A.; Wilson, N.; Merkow, J.; Lan, H.; Marsden, A.; Shadden, S. SimVascular: An open source pipeline for cardiovascular simulation. Ann. Biomed. Eng.
**2017**, 45, 525–541. [Google Scholar] [CrossRef] - Hoi, Y.; Wasserman, B.; Xie, Y.; Najjar, S.; Ferruci, L.; Lakatta, E.; Gerstenblith, G.; Steinman, D. Characterization of volumetric flow rate waveforms at the carotid bifurcations of older adults. Physiol. Meas.
**2010**, 31, 291. [Google Scholar] [CrossRef] [PubMed] - Valen-Sendstad, K.; Piccinelli, M.; KrishnankuttyRema, R.; Steinman, D. Estimation of inlet flow rates for image-based aneurysm CFD models: Where and how to begin? Ann. Biomed. Eng.
**2015**, 43, 1422–1431. [Google Scholar] [CrossRef] [PubMed] - Arzani, A. Coronary artery plaque growth: A two-way coupled shear stress–driven model. Int. J. Numer. Methods Biomed. Eng.
**2020**, 36, e3293. [Google Scholar] [CrossRef] [PubMed] - Kim, H.; Vignon-Clementel, I.; Coogan, J.; Figueroa, C.A.; Jansen, K.; Taylor, C.A. Patient-specific modeling of blood flow and pressure in human coronary arteries. Ann. Biomed. Eng.
**2010**, 38, 3195–3209. [Google Scholar] [CrossRef] - Tu, J.H.; Rowley, C.W.; Luchtenburg, D.M.; Brunton, S.L.; Kutz, J.N. On dynamic mode decomposition: Theory and applications. J. Comput. Dyn.
**2014**, 1, 391–421. [Google Scholar] [CrossRef] [Green Version] - Xiao, N.; Alastruey, J.; Figueroa, C.A. A systematic comparison between 1-D and 3-D hemodynamics in compliant arterial models. Int. J. Numer. Methods Biomed. Eng.
**2014**, 30, 204–231. [Google Scholar] [CrossRef] [Green Version] - Arzani, A.; Shadden, S.C. Wall shear stress fixed points in cardiovascular fluid mechanics. J. Biomech.
**2018**, 73, 145–152. [Google Scholar] [CrossRef] - Chen, K.K.; Tu, J.H.; Rowley, C.W. Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci.
**2012**, 22, 887–915. [Google Scholar] [CrossRef] - Shadden, S.C.; Lekien, F.; Marsden, J.E. Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Phys. D Nonlinear Phenom.
**2005**, 212, 271–304. [Google Scholar] [CrossRef] - Haller, G. Lagrangian coherent structures. Ann. Rev. Fluid Mech.
**2015**, 47, 137–162. [Google Scholar] [CrossRef] [Green Version] - Arzani, A.; Gambaruto, A.M.; Chen, G.; Shadden, S.C. Lagrangian wall shear stress structures and near-wall transport in high-Schmidt-number aneurysmal flows. J. Fluid Mech.
**2016**, 790, 158–172. [Google Scholar] [CrossRef] [Green Version] - Valen-Sendstad, K.; Mardal, K.A.; Steinman, D.A. High-resolution CFD detects high-frequency velocity fluctuations in bifurcation, but not sidewall, aneurysms. J. Biomech.
**2013**, 46, 402–407. [Google Scholar] [CrossRef] [PubMed] - Sieber, M.; Paschereit, C.O.; Oberleithner, K. Spectral proper orthogonal decomposition. J. Fluid Mech.
**2016**, 792, 798–828. [Google Scholar] [CrossRef] [Green Version] - Ghate, A.S.; Towne, A.; Lele, S.K. Broadband reconstruction of inhomogeneous turbulence using spectral proper orthogonal decomposition and Gabor modes. J. Fluid Mech.
**2020**, 888, 1–13. [Google Scholar] [CrossRef] - Scherl, I.; Strom, B.; Shang, J.K.; Williams, O.; Polagye, B.L.; Brunton, S.L. Robust Principal Component Analysis for Modal Decomposition of Corrupt Fluid Flows. arXiv
**2019**, arXiv:1905.07062. [Google Scholar] [CrossRef] - Williams, M.; Kevrekidis, I.; Rowley, C. A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition. J. Nonlinear Sci.
**2015**, 25, 1307–1346. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The three-dimensional models, the corresponding mesh, and flow waveforms used as the inlet boundary condition are shown. The left panel shows the coronary artery stenosis model and the right panel shows the cerebral aneurysm model.

**Figure 2.**The flow waveform divided into different stages is shown for the coronary artery stenosis and cerebral aneurysm model. Each stage is colored differently and the number of snapshots used are shown next to the arrow. The coronary artery flow rate is divided into three stages: low flow rate, flow acceleration, and flow deceleration. The aneurysm’s flow is divided into six stages based on the shape of the waveform: two stages during flow acceleration, three stages during flow deceleration, and one stage during low flow rate in diastole.

**Figure 3.**The relationship between Womersley’s analytical solution and dynamic mode decomposition (DMD) approximation is shown in a rigid tube.

**Top left**: The pressure gradient waveform driving the flow in the rigid tube is shown. Corresponding Fourier series coefficients and frequencies are reported in the table.

**Top right**: The Womersley solution for velocity profile at t = 0.3 s is outlined, and the corresponding coefficients, frequencies, and analytical forms are reported.

**Bottom**: The DMD algorithm is reported. Moreover, the location of discrete eigenvalues in the unit circle is shown and the reconstructed velocity profile based on each eigenvalue is sketched. The assembled velocity profile based on all of the DMD modes and the corresponding frequencies are demonstrated.

**Figure 4.**Two representative velocity DMD modes are shown for the original DMD method applied on the entire cardiac cycle. The location of discrete eigenvalues in the unit circle corresponding to the selected DMD modes are shown. Velocity streamlines are shown and colored by their magnitude. (

**a**) Real part of the DMD mode corresponding to the eigenvalue $\lambda =1$, (

**b**) real part of the DMD mode corresponding to the eigenvalue $\lambda =0.99\pm 0.05i$ for the coronary artery stenosis and (

**c**) real part of the DMD mode corresponding to the eigenvalue $\lambda =1$, (

**d**) real part of the DMD mode corresponding to the eigenvalue $\lambda =0.99\pm 0.10i$ for the cerebral aneurysm model are shown.

**Figure 5.**Velocity and wall shear stress (WSS) computational fluid dynamics (CFD) data and the multistage DMD with control (mDMDc) modes are shown in the first stage of the coronary artery stenosis model. The stage is highlighted with a red dashed line in the flow waveform. The location of discrete eigenvalues in the unit circle corresponding to the selected mDMDc modes are shown. Velocity and WSS streamlines are shown and colored by their magnitude. WSS vectors are normalized and shown on top of the WSS streamlines to show the WSS vector direction. (

**a**) CFD data (time = 0.11 s), (

**b**) reconstructed data with 95% accuracy, (

**c**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.99$, (

**d**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.99\pm 0.04i$, (

**e**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.98\pm 0.08i$.

**Figure 6.**Velocity and WSS CFD data and the mDMDc modes are shown in the second stage of the coronary artery stenosis model. The stage is highlighted with a red dashed line in the flow waveform. The location of discrete eigenvalues in the unit circle corresponding to the selected mDMDc modes are shown. Velocity and WSS streamlines are shown and colored by their magnitude. WSS vectors are normalized and shown on top of the WSS streamlines to show the WSS vector direction. (

**a**) CFD data (time = 0.36 s), (

**b**) reconstructed data with 95% accuracy, (

**c**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =1$, (

**d**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.98\pm 0.15i$, (

**e**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.94\pm 0.03i$.

**Figure 7.**Velocity and WSS CFD data and the mDMDc modes are shown in the third stage of the coronary artery stenosis model. The stage is highlighted with a red dashed line in the flow waveform. The location of discrete eigenvalues in the unit circle corresponding to the selected mDMDc modes are shown. Velocity and WSS streamlines are shown and colored by their magnitude. WSS vectors are normalized and shown on top of the WSS streamlines to show the WSS vector direction. (

**a**) CFD data (time = 0.8 s), (

**b**) reconstructed data with 95% accuracy, (

**c**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =1$, (

**d**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.98\pm 0.06$, (

**e**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.94\pm 0.09i$.

**Figure 8.**Velocity and WSS CFD data and the mDMDc modes are shown in the second stage of the cerebral aneurysm model. The stage is highlighted with a red dashed line in the flow waveform. The location of discrete eigenvalues in the unit circle corresponding to the selected mDMDc modes are shown. Velocity and WSS streamlines are shown and colored by their magnitude. WSS vectors are normalized and shown on top of the WSS streamlines to show the WSS vector direction. (

**a**) CFD data (time = 0.01 s), (

**b**) reconstructed data with 95% accuracy, (

**c**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.99\pm 0.10i$, (

**d**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.98\pm 0.14i$, (

**e**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.97\pm 0.26i$.

**Figure 9.**Velocity and WSS CFD data and the mDMDc modes are shown in the fourth stage of the cerebral aneurysm model. The stage is highlighted with a red dashed line in the flow waveform. The location of discrete eigenvalues in the unit circle corresponding to the selected mDMDc modes are shown. Velocity and WSS streamlines are shown and colored by their magnitude. WSS vectors are normalized and shown on top of the WSS streamlines to show the WSS vector direction. (

**a**) CFD data (time = 0.29 s), (

**b**) reconstructed data with 95% accuracy, (

**c**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.99\pm 0.07i$, (

**d**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.96\pm 0.20i$, (

**e**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.95\pm 0.29i$.

**Figure 10.**Velocity and WSS CFD data and the mDMDc modes are shown in the sixth stage of the cerebral aneurysm model. The stage is highlighted with a red dashed line in the flow waveform. The location of discrete eigenvalues in the unit circle corresponding to the selected mDMDc modes are shown. Velocity and WSS streamlines are shown and colored by their magnitude. WSS vectors are normalized and shown on top of the WSS streamlines to show the WSS vector direction. (

**a**) CFD data (time = 0.47 s), (

**b**) reconstructed data with 95% accuracy, (

**c**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.93\pm 0.10i$, (

**d**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.97\pm 0.17i$, (

**e**) real part of the mDMDc mode corresponding to the eigenvalue $\lambda =0.94\pm 0.17i$.

**Table 1.**The minimum number of modes required to reach a specified reconstruction accuracy in velocity data is shown for the coronary artery stenosis model. The spatiotemporal average of data ($S{T}_{data}$) is shown for each stage.

Accuracy | 90% | 95% | 98% | 99% | 99.5% | $S{T}_{data}$ (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 |

**Table 2.**The minimum number of modes required to reach a specified reconstruction accuracy in wall shear stress (WSS) data is shown for the coronary artery stenosis model. The spatiotemporal average of data ($S{T}_{data}$) is shown for each stage.

Accuracy | 90% | 95% | 98% | 99% | 99.5% | $S{T}_{data}$ (dynes/cm^{2}) | |
---|---|---|---|---|---|---|---|

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 |

**Table 3.**The minimum number of modes required to reach a specified reconstruction accuracy in velocity data is shown for the cerebral aneurysm model. The spatiotemporal average of data ($S{T}_{data}$) is shown for each stage.

Accuracy | 90% | 95% | 98% | 99% | 99.5% | $S{T}_{data}$ (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 |

**Table 4.**The minimum number of modes required to reach a specified reconstruction accuracy in wall shear stress (WSS) data is shown for the cerebral aneurysm model. The spatiotemporal average of data ($S{T}_{data}$) is shown for each stage.

Accuracy | 90% | 95% | 98% | 99% | 99.5% | $S{T}_{data}$ (dynes/cm^{2}) | |
---|---|---|---|---|---|---|---|

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 |

**Table 5.**The minimum number of modes required to reach a specified reconstruction accuracy in velocity data is shown for the traditional DMD model applied to the entire data. The spatiotemporal average of data ($S{T}_{data}$) is reported for each system.

Accuracy | 90% | 95% | 98% | 99% | 99.5% | $S{T}_{data}$ (cm/s) |
---|---|---|---|---|---|---|

Coronary artery stenosis | 27 | 41 | 53 | 77 | 88 | 6.7 |

Cerebral aneurysm | 96 | 110 | 155 | 165 | 173 | 12.1 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Habibi, 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