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REVIEW 3 major objections 5 minor 98 references

Multilevel and multifidelity uncertainty quantification for cardiovascular hemodynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Cheap 0D/1D blood-flow models cut uncertainty-quantification cost by up to 1000x

desk verdict Solid application of MLMF to cardiovascular UQ with real cost gains, but the headline savings are extrapolated from small pilots and need uncertainty bounds before I'd trust the factor-of-10-100 numbers. read the letter →

arxiv 1908.04875 v2 pith:2K2SILQJ submitted 2019-08-13 q-bio.QM physics.comp-phstat.AP

Casey M. Fleeter , Gianluca Geraci , Daniele E. Schiavazzi , Andrew M. Kahn , Alison L. Marsden This is my paper · ORCID
classification q-bio.QMphysics.comp-phstat.AP MSC 65C0592C35
keywords multilevelmultifidelityMonteCarlocardiovascularhemodynamicsuncertaintyquantificationreduced-ordermodelscontrolvariateswallshearstressnormalizedconfidenceintervalpatient-specificmodeling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that uncertainty quantification for patient-specific cardiovascular hemodynamics can be made practical by pooling three model fidelities—full 3D simulations plus 1D and 0D reduced models—inside a single multilevel multifidelity Monte Carlo estimator. The claim is that for the same target confidence interval, this estimator needs one to two orders of magnitude less compute than multilevel Monte Carlo on 3D meshes alone, and one to three orders less than plain Monte Carlo, across healthy and diseased aortic and coronary anatomies. If true, clinical users with constrained computational budgets could report error bars on simulated pressures, flows, and wall shear stress rather than only point predictions.

What carries the argument

The load-bearing object is the multilevel multifidelity (MLMF) estimator, which writes the target expectation as a sum over levels $\ell$ of discrepancies $Y_\ell$ between successive 3D mesh resolutions and, at each level, augments the high-fidelity discrepancy estimator with a control variate $\alpha_\ell(\hat{Y}^{\mathrm{LF}}_\ell-\mathbb{E}[Y^{\mathrm{LF}}_\ell])$ built from a low-fidelity model. The variance reduction is governed by the per-level Pearson correlation $\rho_\ell$ between high- and low-fidelity discrepancies and the cost ratio $w_\ell$; these determine the optimal sample allocation $N_\ell^{\mathrm{HF}}$ and the low-fidelity oversampling factor $r^\star_\ell$. The paper documents high correlations (mostly above 0.99 for global quantities, and above 0.6 for wall shear stress), which is what lets the cheap models carry most of the sampling burden.

What would settle it

Run the 3D-1D-0D workflow to an actual normalized confidence interval of 0.01 for all quantities of interest on a new anatomy, with independent pilot and convergence phases, and compare realized cost with the extrapolated cost; if realized costs systematically exceed the extrapolated costs beyond the spread seen in the paper's single validation case, the central efficiency claim fails in that setting. A cheaper check is to resample the pilot many times with different 25-sample seeds and see whether the extrapolated cost varies by more than an order of magnitude.

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Extended reading notes

Core claim

The central discovery is that the variance-reduction machinery of multilevel Monte Carlo—a telescoping sum of discrepancies between successive mesh resolutions—can be fused with a control-variate correction from cheap low-fidelity models at every level, and for cardiovascular hemodynamics this fusion yields estimators whose normalized confidence interval ($\mathrm{nCI}=6\sigma/\mu$) reaches 0.01 at a fraction of the cost of single-fidelity estimators. With two low-fidelity families (1D and 0D, the latter in full and resistor-only forms) plus three 3D mesh levels, the paper reports extrapolated costs as low as tens of equivalent 3D-fine runs for global quantities, versus thousands for Monte Carlo and hundreds for multilevel Monte Carlo, with local wall-shear-stress quantities showing smaller but still substantial gains. The low-fidelity models need not be unbiased: because each level is corrected by a control variate, bias in the 0D or 1D model is automatically compensated as long as the correlation with the high-fidelity quantity stays high.

Load-bearing premise

The reported cost reductions are extrapolated from variances and correlations estimated in a single pilot run of 25 samples per level, so the savings assume those pilot estimates represent the true sampling behavior of the quantities of interest.

Editorial extensions

If this is right

  • Global quantities of interest (outlet flow, outlet pressure, model-averaged pressure) can be converged to a normalized confidence interval of 0.01 with tens of equivalent 3D-fine runs in the best cases, making routine uncertainty quantification feasible within typical clinical compute budgets.
  • Local quantities such as spatially averaged wall shear stress remain the hard case: even the best MLMF costs are hundreds to thousands of equivalent 3D-fine runs, and diseased geometries widen the gap relative to healthy ones.
  • Adding very cheap 0D models as an extra coarse level improves the 3D-1D-0D scheme over the 3D-1D scheme, with better post-pilot confidence intervals and lower extrapolated costs, most visibly for local quantities.
  • Under a fixed high-fidelity budget equal to 50 fine 3D runs, the MLMF schemes deliver smaller (better) confidence intervals for healthy than for diseased anatomies, and for global than for local quantities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the pilot-based variance and correlation estimates generalize across patient cohorts, the workflow could be wrapped in an adaptive pilot that spends fewer 3D runs per level (e.g., 5-10 instead of 25) and still reaches the target confidence interval, trading expensive high-fidelity samples for many more cheap low-fidelity samples.
  • The method's reliance on correlation suggests a cheap screening test for a new anatomy: measure the per-level correlation between 3D and 1D/0D discrepancies for the target quantity before launching a full campaign; if correlations fall well below the values reported here, the MLMF advantage over multilevel Monte Carlo will shrink accordingly.
  • Time-resolved estimators (full pressure and flow waveforms) would likely preserve the same ordering of gains because the per-time-step variance reduction depends on the same correlations, although the pilot cost would increase by the number of time points reported.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper integrates Sandia Dakota's multilevel multifidelity Monte Carlo (MLMF) estimators with the SimVascular modeling pipeline to perform uncertainty quantification for cardiovascular hemodynamics. Three fidelities are used (3D, 1D, and 0D) with multiple spatial resolution levels, and the framework is demonstrated on healthy and diseased aorto-femoral and coronary anatomies with eight stochastic inputs and 132 or 148 quantities of interest. The central claim is that MLMF estimators achieve one to two orders of magnitude cost reduction over multilevel Monte Carlo and one to three orders over Monte Carlo when extrapolating to a normalized confidence interval of nCI=0.01, with extrapolated costs reported in Tables 5a and 5b. The paper also compares 3D-1D and 3D-1D-0D schemes, healthy versus diseased models, and global versus local QoIs, and it validates the extrapolation procedure for one representative pressure QoI per healthy model in Section 5.2.

Significance. If the reported cost reductions hold, the paper is a valuable demonstration that multilevel multifidelity estimators can make UQ feasible for expensive patient-specific hemodynamic simulations. The work is strengthened by a public workflow repository with a DOI, careful construction and validation of 0D/1D models against 3D results, a wide range of QoIs, and consistent comparisons across methods and anatomies. The main weakness is that the headline cost numbers are extrapolated from small pilot samples, and the validation covers only one pressure QoI per healthy model at variance reductions far smaller than those claimed for the nCI=0.01 target. The manuscript therefore presents a sound methodological application whose quantitative headline needs additional support before the order-of-magnitude claims can be considered robust.

major comments (3)
  1. [§5.1.2, Tables 5a/5b, Eqs. (16), (21), (23)] The extrapolated cost comparisons in Tables 5a and 5b are computed from pilot estimates of level variances, correlations, and cost ratios using only 25 paired samples per level. With n=25, a sample variance has a relative standard error of roughly 28%, and a correlation estimate near 0.8 has a 95% confidence interval of approximately 0.59–0.91; since the control-variate variance reduction factor 1−Λ(r*) depends nonlinearly on ρ² and the cost ratio w, these sampling errors can plausibly change the extrapolated cost ratios by factors of 2–4. No uncertainty or sensitivity analysis is reported for any entry in Tables 5a and 5b. The direct validation in Section 5.2 uses a smaller pilot and targets only variance improvements of ε=0.5, 0.25, and 0.125, which correspond to at most an 8× variance reduction, and it covers one pressure QoI per healthy model rather than the local TAWSS QoIs where the paper still claims 10–100× gains. I request either bootstrap confidence intervals on the extrapolated costs or an independent validation at a variance reduction much closer to the nCI=0.01 target, for several representative QoIs including local WSS.
  2. [§5.2, Eq. (28)] Equation (28) defines the validation target as Vtarget[Q̂_MLMF] = ε Vpilot[Q̂_ML], where Q̂_ML is stated to be the MLMC estimator. If the pilot variance on the right-hand side is not the variance of the MLMF estimator being validated, then the extrapolation exercise does not directly test the MLMF cost curves shown in Figure 10 and Tables 5a/5b. The text should clarify why the MLMC pilot variance is the reference, and if the equation is correct, it should be shown that this target corresponds to the same nCI improvement used in Section 5.1.2. As written, the validation protocol appears to test a different variance-reduction path than the one used for the central cost claims.
  3. [§3.1 and §6 (Discussion)] The manuscript explicitly targets only the variance contribution to the mean squared error and assumes that the 3D model's discretization bias is already satisfactory, with the underlying convergence studies stated to be 'not reported here for brevity'. This is an acceptable choice for a relative cost comparison of estimators, but it means the reported nCI values quantify uncertainty around the 3D model prediction, not around the true clinical quantity. Because the abstract and introduction describe improved accuracy of hemodynamic quantities of interest, the manuscript should state this limitation more prominently and either provide the mesh-convergence data in supplementary material or give a reference where it can be found.
minor comments (5)
  1. [Table 5a] In the Aorto-Femoral Healthy Model TAWSS row, the Monte Carlo effective cost is printed as '23 94.0', which appears to be a typo for '23 940.0'; please correct this.
  2. [Figure 10 caption] The caption says the values shown in green pentagons and triangles are the projected and actual extrapolated costs, respectively, but the legend text is not fully explicit about which symbol corresponds to which quantity; please clarify the symbol mapping.
  3. [§5.2] The validation section uses a pilot with 15, 10, and 5 samples on the three discrepancy levels, whereas the main pilot uses 25 samples per level; the text should state explicitly why this smaller pilot was chosen and how it affects the comparison with the main extrapolation.
  4. [Table 2] Some numbers in Table 2 use inconsistent digit-grouping separators (e.g., '1 219 672', '1 036 483', '1 026,675'); please standardize the formatting.
  5. [§4.2] The phrase 'a simplified LPN was constructed from only resistor elements to serve the purpose of a coarse model representation' is clear, but the later statement that the zero-dimensional models count as 'resolution levels' could be made more explicit about how a lack of spatial resolution is handled in the MLMF hierarchy.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the MLMF estimator is an existing method applied to new cardiovascular UQ problems, and the claimed cost reductions are extrapolated from pilot estimates and partially validated by actual runs, not derived from the target results.

full rationale

The paper's central claim is that MLMF estimators reduce computational cost relative to MC/MLMC for cardiovascular hemodynamics UQ. The method itself is taken from prior work, including self-citations, but the estimator equations are restated in the paper (Eqs. 22-25) and the application is new. The cost reduction figures in Tables 5a/5b are explicitly extrapolated from a 25-sample-per-level pilot run that estimates V[Y_l], rho_l, and per-level costs; this is a standard pilot-extrapolation procedure, not a fit of a parameter that is then renamed a prediction. The paper validates the extrapolation in Section 5.2 with actual simulations for representative QoIs (Figure 10), so the claimed cost curve is checked against measured performance rather than being forced by construction. The self-citations to [46,47,73,74] provide background and the specific MLMF algorithm version, but the load-bearing comparisons are computed from the cardiovascular pilot data and solver costs reported in Tables 2-4, and the QoI values are obtained from direct simulations of the three model fidelities. No uniqueness theorem or ansatz is imported from the authors' prior work to forbid alternatives. The main weakness is statistical: pilot-based estimates of variances and correlations carry sampling error, and the validation covers only one QoI per model at variance reductions up to 8x, but this is an uncertainty/robustness limitation, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's claims rest on standard Monte Carlo theory and on assumptions about the pilot run and high-fidelity bias, which are stated. No invented entities are introduced.

assumptions (4)
  • domain assumption The high-fidelity (3D) model has negligible discretization bias for the quantities of interest.
    Stated in Section 3.1: 'We assume the bias of the high-fidelity model to already be satisfactory.' If false, the reported confidence intervals are overconfident because they only reflect estimator variance.
  • domain assumption The pilot run of 25 samples per level provides reliable estimates of variances, correlations, and costs for extrapolation.
    The extrapolated cost reductions in Tables 5 and 7 depend on these pilot estimates. Validation in Section 5.2 covers only two QoIs, so the reliability for all QoIs is assumed.
  • domain assumption The low-fidelity models are sufficiently correlated with the high-fidelity model at each level for control variates to reduce variance.
    The method requires positive correlation; the paper measures correlations in Table 4, but the values for local TAWSS QoIs are as low as 0.6, which is an assumption that these correlations are high enough for the claimed gains.
  • domain assumption The measured mean computational cost per model level is representative of the cost distribution across the stochastic parameter space.
    Costs in Table 2 are averages from 25 pilot runs with the mean parameter realization, and are used in the extrapolation formulas. Variability in cost across parameter values is not reported.

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Cite this review

Pith. "Pith review of Multilevel and multifidelity uncertainty quantification for cardiovascular hemodynamics." pith.science (2026). https://pith.science/paper/2K2SILQJ

@misc{pith2026190804875,
  author       = {Pith},
  title        = {Pith review of: Multilevel and multifidelity uncertainty quantification for cardiovascular hemodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2K2SILQJ}},
  note         = {Machine review of arXiv:1908.04875}
}
read the original abstract

Standard approaches for uncertainty quantification in cardiovascular modeling pose challenges due to the large number of uncertain inputs and the significant computational cost of realistic three-dimensional simulations. We propose an efficient uncertainty quantification framework utilizing a multilevel multifidelity Monte Carlo estimator to improve the accuracy of hemodynamic quantities of interest while maintaining reasonable computational cost. This is achieved by leveraging three cardiovascular model fidelities, each with varying spatial resolution to rigorously quantify the variability in hemodynamic outputs. We employ two low-fidelity models to construct several different estimators. Our goal is to investigate and compare the efficiency of estimators built from combinations of these low-fidelity and high-fidelity models. We demonstrate this framework on healthy and diseased models of aortic and coronary anatomy, including uncertainties in material property and boundary condition parameters. We seek to demonstrate that for this application it is possible to accelerate the convergence of the estimators by utilizing a MLMF paradigm. Therefore, we compare our approach to Monte Carlo and multilevel Monte Carlo estimators based only on three-dimensional simulations. We demonstrate significant reduction in total computational cost with the MLMF estimators. We also examine the differing properties of the MLMF estimators in healthy versus diseased models, as well as global versus local quantities of interest. As expected, global quantities and healthy models show larger reductions than local quantities and diseased model, as the latter rely more heavily on the highest fidelity model evaluations. In all cases, our workflow coupling Dakota's MLMF estimators with the SimVascular cardiovascular modeling framework makes uncertainty quantification feasible for constrained computational budgets.

Figures

Figures reproduced from arXiv: 1908.04875 by the authors.

Figure 1
Figure 1. Improved fine-scale resolution for quantities of interest comes at the expense of higher computational cost when [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Agreement for outlet flow and pressure quantities of interest is seen between 3D, 1D, and 0D models with resistance [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Multilevel multifidelity simulations are comprised of (a) a multifidelity control variate approach coupling high and [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: The developed workflow utilizes Sandia National Laboratories’ Dakota toolkit to automate the uncertainty quantifi [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Healthy and diseased aorto-femoral and coronary models of three varying fidelities are used in this study. Specifically, [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Sample convergence plot for mesh selection. Selected meshes for aorto-femoral diseased model demar￾cated with red xs. (For interpretation of the references to color in this figure leg￾end, the reader is referred to the web ver￾sion of this article.) Aorto-Femoral Aorto…
Figure 7
Figure 7. Figure 7: Examining the bias between MC estimators for different model fidelities for (a) a global flow QoI and (b) a local [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Comparing the performance of estimators from MC, MLMC, and MLMF UQ schemes for (top) aorto-femoral and [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: Comparing the performance of UQ estimators from four methods for global and local QoIs. The four methods are [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 10
Figure 10. Figure 10: Validating the extrapolation of estimators for the MLMF UQ schemes for (a) aorto-femoral and (b) coronary healthy [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 11
Figure 11. Figure 11: Comparing the performance of MLMF estimators from 3D and 1D models to the MLMF estimators from 3D, 1D, [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]
Figure 12
Figure 12. Figure 12: Comparison of estimators for healthy and diseased model QoIs. Model is shown with strips of interest colored by [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]
Figure 13
Figure 13. Figure 13: Comparison of estimators for healthy and diseased model QoIs. Model is shown with strips of interest colored by [PITH_FULL_IMAGE:figures/full_fig_p029_13.png]
Figure 14
Figure 14. Figure 14: Comparing the performance of 3D-1D-0D MLMF estimators from healthy and diseased geometries for the (top) [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]
Figure 15
Figure 15. Figure 15: Comparison of estimators for global and local model QoIs for the (top) aorto-femoral and (bottom) coronary models. [PITH_FULL_IMAGE:figures/full_fig_p031_15.png]

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