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Data-driven cardiovascular flow modeling: examples and opportunities

Amirhossein Arzani, Scott T. M. Dawson

arXiv:2010.00131v2physics.comp-phphysics.flu-dyn

TL;DR

Cardiovascular flow modeling is constrained by parameter uncertainty, limited-resolution sampling, and noisy measurements. This paper reviews data-driven techniques and illustrates their use in cardiovascular fluid mechanics, showing how they may support data interpretation and modeling.

  • Problem

    Cardiovascular flow investigations face parameter uncertainty, limited-resolution sampling, and noisy measurements, while high-resolution data are needed for tasks such as setting inlet waveforms and calculating wall shear stress.

    Source: §1 · p. 2, §4.1 · p. 10, §5.4 · p. 16

  • Method

    The paper reviews data-driven modeling techniques and provides examples of their potential use in cardiovascular fluid mechanics.

    Source: §1 · p. 3

  • Results

    The examples illustrate how these techniques could facilitate data interpretation, augment incomplete or low-resolution data, improve data quality, and leverage multiple modalities.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Familiarity with the underlying mathematical tools may help cardiovascular researchers develop customized techniques to improve their models.

    Source: §1 · p. 3

    Robust PCA requires corrupted data to be sparsely distributed, constraining its use when corruption is not sparse.

    Source: §3.4 · p. 10

Abstract

from arXiv · show

High-fidelity modeling of blood flow is crucial for enhancing our understanding of cardiovascular disease. Despite significant advances in computational and experimental characterization of blood flow, the knowledge that we can acquire from such investigations remains limited by the presence of uncertainty in parameters, low spatiotemporal resolution, and measurement noise. Additionally, extracting useful information from these datasets is challenging. Data-driven modeling techniques have the potential to overcome these challenges and transform cardiovascular flow modeling. In this paper, we review several data-driven modeling techniques, highlight the common ideas and principles that emerge across numerous such techniques, and provide illustrative examples of how they could be used in the context of cardiovascular fluid mechanics. In particular, we discuss principal component analysis (PCA), robust PCA, compressed sensing, the Kalman filter for data assimilation, low-rank data recovery, and several additional methods for reduced-order modeling for cardiovascular flows, including the dynamic mode decomposition (DMD), and the sparse identification of nonlinear dynamics (SINDy). All of these techniques are presented in the context of cardiovascular flows with simple examples. These data-driven modeling techniques have the potential to transform computational and experimental cardiovascular flow research, and we discuss challenges and opportunities in applying these techniques in the field, looking ultimately towards data-driven patient-specific blood flow modeling.

1 Introduction

Cardiovascular flow data from CFD, MRI, and laboratory measurements contain uncertainty, artifacts, limited resolution, and noise, while extracting useful information remains challenging. This paper introduces data-driven modeling approaches and examples for interpreting, augmenting, and improving such data, with an emphasis on low-dimensional representations and sparsity.

1 Introduction

  • CFD, 4D flow MRI, PIV, and PTV provide time-resolved blood-flow data, but face numerical error, parameter uncertainty, imaging artifacts, low resolution, and measurement noise.The introduction asks how to improve data quality, combine modalities, extract useful information, and simplify complex spatiotemporal patterns.Source: §1 · p. 2
  • The reviewed methods exploit low-dimensionality and sparsity to represent complex, high-dimensional data compactly and potentially handle corrupted or under-resolved measurements.The authors contrast these methods with deep learning, noting their use with small, sometimes corrupted datasets and their potential to facilitate physical interpretation.Source: §1 · p. 2
  • Low-rank data can be represented by a small basis, enabling sparse representations for reconstructing full states from limited sensor measurements and identifying sparse governing equations.The manuscript proceeds through technique motivation, theory, cardiovascular examples, and opportunities and challenges.Source: §1 · pp. 2–3, §1.1 · p. 5
  • The review surveys data-driven techniques and cardiovascular examples aimed at interpreting data, augmenting incomplete or low-resolution data, improving data quality, and combining modalities.The authors also recommend mathematical references and organize the manuscript around technique motivation, theory, examples, and opportunities and challenges.Source: §1 · p. 3, §1.1 · p. 5

2 Principal component analysis (PCA): detecting redundancy and low-dimensionality in data

PCA uses correlated flow snapshots to identify a compact set of modes that captures much of the data. In aneurysm examples, fewer modes capture 90% of cerebral than abdominal aortic aneurysm data, while noise, sampling, and linearity constrain interpretation.

2.1 Motivation

  • PCA uses singular value decomposition to find an optimal basis of minimally correlated modes, reducing redundancy among correlated spatial or temporal snapshots.Modes are spatial functions; their singular values indicate how much they characterize the data, and time-resolved coefficients describe their snapshot amplitudes.Source: §2.1 · pp. 5–6, §2.2 · p. 7

2.3 Example: Low-dimensional behavior in cerebral and aortic aneurysm hemodynamics

  • Six modes capture 90% of cerebral aneurysm data, versus 16 for abdominal aortic aneurysm data; the authors attribute the difference to more complex, chaotic aortic flow.One cerebral mode captures 70% of the data, and the mode count balances model complexity against fidelity.Source: §2.3 · p. 7
  • The example applies PCA to 50 time steps of patient-specific CFD velocity data from cerebral and abdominal aortic aneurysms.The section frames the comparison around low-dimensionality and the number of modes needed to reconstruct each dataset.Source: §2.3 · p. 7

2.4 Opportunities and challenges

  • PCA can also preprocess data-driven models and reveal correlations across spatial locations or parameter-varying datasets, but requires consistent snapshot alignment and cannot capture nonlinear patterns.The authors identify manifold learning and nonlinear PCA extensions as directions for future work.Source: §2.4 · pp. 7–8
  • PCA results depend on adequate, representative data, and noisy or biased samples can affect convergence; adding data should not materially change the analysis once it converges.The required quantity and quality also depend on the number of modes, their energy, and the accuracy needed for subsequent analysis.Source: §2.4 · pp. 8–9

3 Robust principal component analysis (RPCA): noisy and fluctuating data

Robust PCA separates correlated flow data from sparse corruption, addressing noise that can obscure low-dimensional structure in ordinary PCA. In a noisy aneurysm example, RPCA reveals low-dimensionality, but relies on sparse corruption and requires sensitivity to its regularization parameter to be assessed.

3.1 Motivation

  • Noise and fluctuations can inflate PCA mode counts and obscure mean-flow temporal correlations, motivating RPCA's separation of sparse corruption from the remaining data.The concern applies to experimental measurements and can also arise in computational data, including transitional aneurysm flow.Source: §3.1 · p. 9

3.2 Model theory and background

  • RPCA models the data as a low-rank correlated component plus a sparse component for noisy, fluctuating, or corrupted entries.Its optimization relaxes rank and sparsity penalties to convex nuclear- and l1-norm terms, with λ controlling sparsity; PCA can then be applied to the recovered low-rank component.Source: §3.2 · p. 9

3.3 Example: Low-dimensional behavior in hemodynamics data with noise

  • With noise added to 30% of AAA velocity data, ordinary PCA fails to reveal low-dimensionality, whereas RPCA does; RPCA on clean data also yields fewer dominant modes by assigning some data to the discarded sparse component.The added noise is zero-mean Gaussian with standard deviation 10% of maximum streamwise velocity.Source: §3.3 · p. 10

3.4 Opportunities and challenges

  • RPCA's practical use requires corrupt data to be sparsely distributed, and its sensitivity to λ remains to be investigated for cardiovascular flows.The authors identify experimental hemodynamics as a potential application and suggest RPCA may also support noise-sensitive methods such as DMD.Source: §3.4 · p. 10

4 Compressed sensing (CS): Reconstructing high-resolution data from low-resolution sampling

Compressed sensing (CS) uses known sparsity in a basis to reconstruct high-resolution flow data from low-resolution measurements. A waveform example recovers major features but retains artificial high-frequency components, and practical use depends on suitable sparsifying bases and sampling.

4.1 Motivation

  • CS addresses limited experimental sampling by reconstructing high-resolution data from low-resolution measurements when the data are sparse in a known basis.The paper illustrates a possible spatial reconstruction from typical 3 mm 4D flow MRI sampling to 0.5 mm CFD resolution.Source: §4.1 · p. 10

4.2 Model theory and background

  • The reconstruction represents high-resolution data as x = Φs and uses an l1-regularized optimization to find a sparse vector consistent with measurements.The l1 norm serves as a computationally tractable convex proxy for the l0 count of nonzero coefficients; λ controls sparsity and permits imperfect measurement fit.Source: §4.2 · p. 12

4.3 Example: High-resolution reconstruction of blood flow waveform from low-resolution data

  • From 100 samples across 20 cardiac cycles, CS reconstructs a 1000-point aortic waveform with recovered features but artificial high-frequency components that Gaussian smoothing may remove.The example assumes sparsity in the discrete cosine transform basis.Source: §4.3 · p. 12, Figure 3 · p. 13

4.4 Opportunities and challenges

  • Successful reconstruction requires a suitable sparsifying basis and sampling incoherent with it; random sampling can provide incoherence, whereas uniform sampling conflicts with a Fourier basis.Dictionary or transform learning and training data are discussed as possible ways to identify an appropriate basis, but success for complex blood flow remains to be investigated.Source: §4.4 · p. 13

5 Data assimilation using the Kalman filter: Merging experimental and computational data

Data assimilation combines computational predictions with experimental observations to improve solution reliability despite errors in both. In a transient vortex example, Kalman filtering improves trajectory estimates, while the authors note that error statistics may be uncertain and non-Gaussian.

5.1 Motivation

  • Data assimilation advances a computational model in time and adjusts its predictions using available experimental observations, which may themselves be low-resolution and noisy.Computational models also have limitations from parameter and governing-equation uncertainty.Source: §5.1 · p. 14

5.2 Model theory and background

  • The Kalman filter weights model predictions and measurements according to their expected errors, assuming Gaussian error distributions.Measurements need not be available at every model timestep, and the prediction update is applied when observations are available.Source: §5.2 · p. 14

5.3 Example: Merging computational and experimental trajectories in transient vortical flows

  • In the transient Hill’s vortex example, the Kalman filter significantly outperforms a perturbed-initial-condition trajectory, though noise remains in the filtered estimate.The example uses noisy synthetic observations at a lower sampling rate than the computational model.Source: §5.3 · p. 15, Figure 4 · p. 16

5.4 Opportunities and challenges

  • Kalman-filter data assimilation has been applied to cardiovascular model estimation, but error magnitudes may need estimation and Gaussian errors may be an unsuitable assumption.The authors identify multi-modality assimilation and comparison of methods as opportunities for future work.Source: §5.4 · pp. 15–16

6 Reduced-order physics: Dynamic mode decomposition (DMD) and proper orthogonal decomposition (POD)

POD and DMD reduce flow data to spatial modes and temporal coefficients, but emphasize different properties: reconstruction energy for POD and dynamics for DMD. An aneurysm example shows similar leading modes and different structures among later modes, while long-term extrapolation remains challenging.

6.1 Motivation

  • POD and DMD represent flow evolution through coherent modes, supporting flow-physics analysis and reduced-order modeling.POD ranks modes by energy and optimizes reconstruction, whereas DMD isolates dynamical features; POD’s frequency mixing can reduce physical interpretability.Source: §6.1 · pp. 16–17

6.2 Model theory and background

  • POD uses the SVD to obtain orthogonal spatial modes and temporal coefficients, while DMD fits a linear operator advancing time-resolved snapshots by one timestep.For DMD, mode eigenvalues encode growth or decay and oscillation frequency; the truncation parameter controls the number of modes.Source: §6.2 · pp. 17–18

6.3 Example: Blood flow physics in a cerebral aneurysm using DMD and POD

  • In the idealized cerebral aneurysm example, the leading POD and DMD modes are similar, while later POD modes are more complex and DMD modes reflect the dominant vortex.The comparison uses the final of three simulated cardiac cycles.Source: §6.3 · p. 18, Figure 5 · p. 19

6.4 Opportunities and challenges

  • Long-term extrapolation beyond training data remains challenging, and pulsatile flow variability across cardiac-cycle phases complicates direct application of modal methods.Multistage DMD with control has been proposed to address challenges in blood-flow physics analysis with DMD.Source: §6.4 · pp. 18–19

7 Machine learning reduced-order models (ROM): Overcoming uncertainty in computational models using low-fidelity experimental data

The section presents reduced-order modeling as a way to combine high-resolution simulations across uncertain parameters with low-resolution measurements. Its examples reconstruct flow profiles and identify unknown parameters, while the discussion notes practical challenges in merging CFD and MRI data.

7.1 Motivation

  • Patient-specific CFD parameters such as inlet waveform, flow split, rheology, geometry, and vessel-wall properties may be uncertain, while 4D flow MRI is limited by resolution and noise.The framework is proposed to combine high-resolution computational simulations with low-resolution experimental data to address these limitations.Source: §7.1 · p. 19

7.2 Model theory and background

  • The framework builds a library of parameter-specific POD modes, fits their coefficients to low-resolution measurements with compressed sensing, and reconstructs high-resolution data from the active modes.The library represents possible system dynamics across the considered parameter range; random measurement locations are recommended for compressed sensing.Source: §7.2 · p. 20

7.3 Example 1: Reconstruct high-resolution Womersley flow with uncertain Womersley number and optimal sensor placement

  • With five radial samples, optimal placement reconstructed Womersley profiles with l2 errors of 0.002 for clean data and 0.03 for noisy data, outperforming random placement.Random placement produced l2 errors of 0.35 for clean data and 0.45 for noisy data; optimal placement was more robust to noise.Source: §7.3 · p. 21, Figure 6 · p. 22

7.4 Example 2: Reconstruct high-resolution cerebral aneurysm flow with uncertain viscosity

  • Using one snapshot and 150 random sensors, the method identified the groundtruth viscosity, µ=0.2 P, and reconstructed velocity patterns close to the original.With 30 sensors it still identified the viscosity, though reconstruction accuracy was reduced.Source: §7.4 · p. 22, Figure 7 · p. 23

7.5 Opportunities and challenges

  • Merging CFD and 4D flow MRI faces geometry discrepancies and the computational expense of building the simulation library; parameterized geometries and greedy selection are proposed as possible mitigations.The framework could also be extended to multiple uncertain parameters.Source: §7.5 · pp. 22–23

8 Low-rank data recovery from random spatiotemporal measurements

Low-rank matrix recovery is presented as a way to reconstruct hemodynamics from randomly sampled space-time measurements without specifying a sparse basis. The examples show that reconstruction is less accurate in complex flow regions and that the method requires random sampling.

8.1 Motivation

  • Low-rank recovery may reconstruct full hemodynamics data from partial observations even when noise makes the matrix full-rank, provided it exhibits low-rank behavior under a relaxed rank threshold.Temporal snapshots in complex flows may be highly correlated without being perfectly correlated.Source: §8.1 · pp. 23–24

8.2 Model theory and background

  • The method seeks a low-rank matrix matching observed entries, then relaxes the computationally difficult optimization to a convex nuclear-norm problem solvable by singular value shrinkage.The observed entries correspond to random spatial and temporal samples.Source: §8.2 · p. 24

8.3 Example: Reconstruct high-resolution blood flow with random spatiotemporal measurements

  • At 40% random spatiotemporal sampling, velocity outside the aneurysm was captured perfectly, with mostly low error inside the aneurysm and coronary stenosis; at 10%, complex regions were reconstructed less well.At 10% sampling, errors near peak flow remained for the point outside the aneurysm.Source: §8.3 · p. 24, Figure 8 · p. 25

8.4 Opportunities and challenges

  • Incomplete measurements must be random in space and time; converting uniform samples to random samples requires careful design to avoid accuracy loss.Matrix completion is presented as useful when measurements are missing and the data are highly correlated.Source: §8.4 · p. 25

9 Sparse identification of nonlinear dynamics (SINDy): Discovering analytical dynamics

SINDy uses sparsity in a candidate-function library to identify governing dynamics from data. Examples recover thrombosis reaction equations and time-dependent vortex velocity terms, while highlighting conditioning and library-selection challenges.

9.1 Motivation

  • SINDy assumes governing equations can be selected sparsely from an appropriately chosen candidate-function set [7].Its goal is to approximate system dynamics analytically using a few functions.Source: §9.1 · p. 26

9.2 Model theory and background

  • The method represents state derivatives using candidate functions and sparse coefficients, with sequential thresholded least squares proposed for noisy data.The library can include linear, quadratic, and other functions; the coefficient vectors select active terms.Source: §9.2 · p. 26

9.3 Example 1: Discover a blood coagulation and thrombosis model

  • For the Papadopoulos thrombosis model, SINDy exactly captured active terms and coefficients using solution trajectories generated from the reduced model, whose library included linear and quadratic functions of state variables.Source: §9.3 · p. 27

9.4 Example 2: Discover analytical velocity in transient vortical flows

  • With prior knowledge of ω, SINDYc accurately predicted the active velocity terms and coefficients from a simulated particle trajectory in an unsteady vortex.The candidate library included spatial linear and quadratic functions and time-dependent sine and cosine terms.Source: §9.4 · p. 28

9.5 Opportunities and challenges

  • SINDy may derive reaction-kinetics models or analytical velocity fields, but correlated trajectories can ill-condition Θ, and omitted active terms can produce dense solutions.Sparse representation may not be possible for every system; the vortex example is idealized, and applicability to left-ventricle flows remains to be investigated.Source: §9.5 · p. 28

10 Other techniques and applications

The section surveys additional data-driven applications, including SINDYc, machine learning, physics-informed neural networks, parametric reduced-order models, and multiscale modeling. It also identifies data and parameter-variation challenges that bound these opportunities.

SINDy with control

  • SINDYc exactly captures the correct terms for the velocity vector field of an unsteady Hill’s spherical vortex from a particle trajectory.Figure 10 lists the identified terms for the field.Source: Figure 10 · p. 29

Machine learning and neural networks

  • Neural networks can fit clinical outcomes from simulation inputs and learn turbulence models, while PINNs incorporate governing equations and have been used for inverse velocity estimation and 4D flow MRI denoising.Learning spatially and temporally varying outputs such as velocity or wall shear stress requires extremely large datasets, and near-term feasibility is unclear.Source: §10 · pp. 29–30

Parametric reduced-order models

  • Parametric reduced-order models address the limitation that fixed-parameter Galerkin-POD models cannot predict when model parameters change.They may be valuable when blood-flow or structural-mechanics parameters vary during disease-growth simulations.Source: §10 · p. 30

Data-driven multiscale modeling

  • Reduced-order models can accelerate multiscale simulations, while data-driven models could use simulation or experimental data to approximate underlying physics.The section identifies incomplete biological data for parameter calibration and multimodality integration as challenges, and presents hybrid modeling as a potential route toward digital twins.Source: §10 · p. 30

11 Conclusion

The conclusion characterizes data-driven cardiovascular flow modeling as an early-stage field with opportunities for customized models. It highlights applications spanning data quality, reconstruction, assimilation, parameter identification, and reduced-order modeling.

11 Conclusion

  • Data-driven cardiovascular flow modeling remains in its infancy, with opportunities to develop customized models for cardiovascular biomechanics research.The conclusion highlights multimodality and multifidelity approaches among these opportunities.Source: §11 · p. 30
  • Potential applications include data assimilation, improved data quality, parameter identification, super-resolution flow reconstruction, and reduced-order modeling.The authors hope the review inspires models addressing outstanding challenges in cardiovascular biomechanics.Source: §11 · p. 31

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