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REVIEW 4 major objections 5 minor 52 references

Robust Monte-Carlo Simulations in Diffusion-MRI: Effect of the substrate complexity and parameter choice on the reproducibility of results

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Monte-Carlo diffusion-MRI simulations require at least 500,000 particles and 10,000 cylinders before their ground-truth signals become reproducible.

desk verdict A solid calibration paper with a useful new substrate-generation framework; the headline thresholds are provisional since no threshold criterion is defined, but the direction of each finding checks out and the bootstrapped first study is well done. read the letter →

arxiv 1908.11203 v2 pith:4LNC6ZPD submitted 2019-08-29 physics.med-ph eess.SP

Jonathan Rafael-Patino , David Romascano , Alonso Ramirez-Manzanares , Erick Jorge Canales-Rodríguez , Gabriel Girard , Jean-Philippe Thiran This is my paper · ORCID
classification physics.med-pheess.SP
keywords Monte-Carlodiffusionsimulationdiffusion-weightedMRImicrostructureimagingreproducibilityaxonalundulationsubstrategenerationextra-axonalbiasgroundtruthvalidation
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

Monte-Carlo simulations are widely used to generate ground-truth diffusion-MRI signals for testing microstructure models, but the cheap settings many studies use may not be safe. This paper argues that below $5\times10^5$ particles and $1\times10^4$ time steps, the simulated signal carries significant variance in both the compartment inside the axons and the space outside them, and that substrates built from fewer than 10,000 cylinders produce an extra-axonal signal that is no longer directionally symmetric. It also shows that representing axons as straight cylinders hides a real diameter-estimation error once undulation is introduced, and it presents a substrate-generation framework that builds complex, non-overlapping crossing fibre configurations while preserving volume and packing density. If these claims hold, past and future Monte-Carlo validations should report particle counts, step counts, and substrate sizes, and rethink what a reliable ground truth requires.

What carries the argument

The argument rests on two mechanisms. The first is a bootstrapped error analysis: repeated Monte-Carlo runs at each particle and step count are compared with a high-quality reference, namely the analytical Gaussian-phase-distribution solution for restricted diffusion in cylinders for the intra-axonal signal and a 20-million-particle Monte-Carlo run for the extra-axonal signal, using the Relative Mean Absolute Error (RMAE) to turn raw signal deviations into convergence curves that set the recommended minima. The second is an energy-optimization substrate generator: an objective function penalizes overlap, curvature, and length of parametrized strands, and a modified cylinder-cylinder collision term lets those strands be subdivided into gamma-distributed diameters so that crossing bundles interdigitate without intersecting, preserving the volume fraction in the crossing area.

What would settle it

Recompute the extra-axonal signal for the same substrate with an independent finite-element or analytical effective-medium solver across the full acquisition protocol; if the reference signal disagrees by more than the reported 0.4–0.7% convergence margin, the recommended parameter minima are miscalibrated.

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Extended reading notes

Core claim

On its own terms, the paper establishes three quantitative claims. First, for the substrate and acquisition settings studied, simulations with fewer than $5\times10^5$ particles or $1\times10^4$ steps show a significant, avoidable variance between repeated runs in both the restricted intra-axonal signal and the hindered extra-axonal signal; the mean relative absolute error settles near 0.4–0.7% only above those values. Second, substrates built from fewer than 10,000 sampled cylinders produce extra-axonal radial signals whose directional symmetry is visibly broken and whose mean amplitude is biased, with the bias disappearing around a 200–230 µm voxel scale for the tested diameter distribution. Third, replacing straight cylinders with helically undulating axons shifts the apparent diameter recovered from a straight-cylinder model, severely for 1 µm axons, where the mis-estimation can exceed 300%. The paper also presents a framework for generating complex crossing substrates whose strands interdigitate without overlap, preserving volume in the crossing region and achieving an intra-axonal volume fraction above 48% even at low resolution.

Load-bearing premise

The load-bearing premise is that the 20-million-particle, $2\times10^4$-step extra-axonal run is itself an accurate reference; if that Monte-Carlo simulation carries a systematic bias from the fixed step size, substrate representation, or cylinder packing, then every error percentage and the recommended $5\times10^5$-particle, $1\times10^4$-step minimum inherit that bias.

Editorial extensions

If this is right

  • Validations run below $5\times10^5$ particles and $1\times10^4$ steps should be treated as carrying unexplained run-to-run variance; reporting a single run is not enough.
  • Extra-axonal signals from substrates with fewer than about 10,000 cylinders are not radially isotropic, so fitting procedures that assume axial symmetry inherit a systematic bias.
  • Axon-diameter estimates from straight-cylinder models can be off by more than 300% for 1 µm undulating axons, meaning undulation is a confounding factor in diameter mapping.
  • The proposed substrate generator can produce crossing-fibre ground truths with preserved volume and an intra-axonal volume fraction above 48% at coarse resolution, giving crossing-aware models a harder, more realistic test.

Reading between the lines

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

  • A practical next step the paper does not take is to derive a scaling law for the particle and step minima as a function of diffusivity, diffusion time, and packing density, since the recommended values are tied to the specific protocol and substrate studied here.
  • Because the extra-axonal reference is itself a Monte-Carlo run, an independent finite-element or effective-medium calculation on the same substrate would give a stronger absolute calibration of the recommended thresholds.
  • The generator could be used to build controlled dose-response phantoms that vary undulation amplitude, crossing angle, and volume fraction independently, letting models be scored against known ground truth rather than inferred tissue properties.
  • The observed substrate-size bias in the mean extra-axonal signal implies that shrinking the simulated voxel is not a neutral computational shortcut, and time-dependent diffusion metrics extracted from such simulations may inherit a false time-dependence.
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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

4 major / 5 minor

Summary. This manuscript studies three methodological pitfalls in Monte-Carlo diffusion simulations (MCDS) used as ground truth for diffusion MRI: the number of particles and time steps, the intra-axonal representation (straight vs. undulating cylinders), and the size of the extra-axonal substrate. It reports that simulations with fewer than 5e5 particles and 1e4 steps show significant variability, and that substrates with fewer than 10,000 sampled cylinders induce a bias in the radial symmetry of the extra-axonal signal. It then presents a framework for generating complex crossing substrates that preserves volume in the crossing region and achieves high packing density, and it evaluates axon diameter estimation in such substrates. The paper also provides a complexity analysis of the simulator and makes the code and substrate data available.

Significance. If the central quantitative claims hold, the paper is practically important because many published MCDS-based ground-truth studies use parameter values below the proposed minima, and the paper identifies a concrete reproducibility risk. The bootstrap design in Section 3.1, with 50 repetitions per parameter combination, is a clear strength, as is the effort to validate the simulator against analytical solutions and an independent FEM approach. The proposed crossing-substrate framework is a useful step toward more realistic numerical phantoms, and the volume-preservation property is a genuine improvement over naive crossings. However, the paper's headline numbers (5e5 particles, 1e4 steps, 10,000 cylinders) are currently derived from visual inspection without a pre-specified tolerance, and the extra-axonal gold standard is itself a Monte-Carlo simulation whose systematic errors have not been independently bounded. These issues make the main recommendations not yet fully reproducible or falsifiable as stated, so the manuscript needs substantial revision before it can serve as a reliable reference for parameter choice.

major comments (4)
  1. [§4.1, §6] The central thresholds '5×10^5 particles' and '1×10^4 steps' are presented in the conclusions without a pre-specified acceptance criterion. In Figures 5 and 6 the RMAE is plotted against sample size, but no tolerance on the RMAE is defined and no confidence bounds are derived from the 50 repetitions; the reader cannot determine why 5×10^5 rather than 2×10^5 or 1×10^6 is the cut-off. In addition, when the number of steps is varied the reference uses the maximum value (2×10^4), so the RMAE for smaller step counts mixes the fixed-step discretization bias with Monte Carlo variance, and the text labels the combined effect 'variance'. Please define an explicit reproducibility criterion (e.g., a maximal acceptable RMAE with a confidence interval) and separate the discretization bias from the sampling variance, ideally by varying the step count in the gold-standard reference as well.
  2. [§3.1, §4.1] The extra-axonal gold standard is itself a Monte-Carlo simulation (20×10^6 particles, 2×10^4 steps), and the paper reports no comparison of this reference against an independent method beyond an assertion of convergence; the earlier FEM validation [38] is cited but not shown for this substrate and protocol. If the fixed-step integration or the cylinder packing carries a systematic bias, every RMAE in Section 4.1 inherits it, so the recommended minima could be miscalibrated. Please add either a convergence study for the extra-axonal gold standard with respect to step size (not just particle number) or a comparison with an independent numerical/analytical reference for the identical geometry.
  3. [§3.3, §4.3] The conclusion that substrates with fewer than 10,000 cylinders induce an 'important bias' is not supported by a quantitative threshold. The anisotropy metric (std/mean of the radial signal) is reported for one random substrate per size, with only sizes 100, 1,000, 10,000, 50,000, and 100,000 tested; the boundary 'less than 10,000' is an interpolation with no uncertainty. Please repeat the measurement over multiple random packing realizations, report the spread of the anisotropy metric, and state a pre-defined isotropy criterion (or at least a tolerance on the deviation from the large-substrate limit).
  4. [§3.2, §4.2] The diameter-fitting intervals in Figure 7 depend on the arbitrary threshold of '1% difference from the minimum fitting error'. Because the RMAE surface is strongly protocol-dependent (Figure 8), the reported intervals and the qualitative claim of 'considerable mis-estimation' are sensitive to this ad hoc choice. Please justify the 1% level (e.g., by linking it to the noise level or the known sensitivity of the protocol) or report how the intervals change when the threshold is varied.
minor comments (5)
  1. [§3.1] The list of tested particle numbers includes 2×10^6, but the preceding sentence says particles were varied from 1×10^3 to 1×10^6; please make the range consistent.
  2. [§3.2] The sentence 'choosing a the parameters that shows almost almost no variance' contains repeated words ('a the', 'almost almost') and a subject-verb agreement error; please correct it.
  3. [§3.3] 'This parameters where chooseng from the previous results' should read 'These parameters were chosen from the previous results'.
  4. [§4.1, Figure 5] The caption says 'RMAE of all the repetition' but should read 'RMAE of all the repetitions'; please also indicate which panels correspond to the intra- and extra-axonal cases in the text, since the figure has six panels.
  5. [§3.4] The optimization time (about 42 hours) and simulation time (less than 24 hours) are reported for a single run; please state whether these are wall-clock times and on what specific hardware configuration they were obtained, to make the computational claims reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: thresholds and evaluations are empirical measurements against independent analytical and FEM references, not fitted re-statements of inputs.

full rationale

The paper's central quantitative claims are descriptive summaries of its own Monte-Carlo convergence and substrate-size experiments, not quantities derived from a fitted model that is then re-predicted. Intra-axonal signals are compared against the independent GPD analytical solution (Van Gelderen et al., [48]) and extra-axonal signals against a high-particle gold standard that was additionally checked against a finite-element method in [38]; even though [38] shares an author, it is an independent numerical method and the paper also reports its own convergence check at 1e6 particles/5e3 steps, so the citation is not load-bearing. The recommended thresholds (5e5 particles, 1e4 steps, 10,000 cylinders) are read off RMAE and anisotropy plots, with no equation forcing those numbers by construction; the absence of a formal threshold criterion is a reproducibility weakness, not a circularity. The undulation/diameter study simulates known geometries and fits them with a deliberately misspecified straight-cylinder GPD model, so the mis-estimation is a model-error measurement rather than an input re-stated as an output. The framework evaluation uses the same simulator but tests geometric properties and convergence, with no fitted parameter renamed as a prediction. No equation in the paper reduces to its own input, and no load-bearing claim rests on a self-citation chain.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper's conclusions rest on the fidelity of the simulation engine, the GPD reference, the representativeness of the random packing, and tuning parameters (w_o, w_c, w_l) that are never disclosed. These are not free physical parameters in a theory, but they are free parameters of the computational pipeline.

free parameters (2)
  • Cost-function weights w_o, w_c, w_l = not specified
    Equation 8 defines the substrate optimization cost but never gives the weight values; these are hand-chosen and load-bearing for the claimed non-overlap, volume preservation, and packing density.
  • Diameter fitting threshold (1% above minimum RMAE) = 1%
    Chosen in Section 3.2 to define the plausible diameter range; this threshold affects all reported diameter mis-estimation intervals.
assumptions (4)
  • domain assumption The Gaussian Phase Distribution (GPD) analytical solution for restricted diffusion in straight cylinders is the correct reference for intra-axonal signals.
    Used as ground truth in Sections 3.1 and 3.2 for the intra-axonal compartment; GPD is an approximation for small radii and short times and is not exact, so errors relative to it are not absolute.
  • domain assumption The cylinder packing algorithm from [22] produces an extra-axonal space representative of white matter.
    The extra-axonal signal and the anisotropy findings in Section 4.3 depend on the random packing geometry being biomimetic; the paper does not validate against tissue histology.
  • domain assumption The fixed-step-size random walk with r = sqrt(6D dt) accurately models Brownian motion in these restricted geometries.
    Adopted from [22, 7] in Section 2.1; the collision handling and step discretization could introduce systematic bias, especially at high b-values.
  • domain assumption The simulator itself is correct because it was validated against analytical solutions and a finite element method in [38].
    The validation is cited to a conference abstract with overlapping authors; the present paper does not reproduce that validation, so it is a background trust assumption.

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Cite this review

Pith. "Pith review of Robust Monte-Carlo Simulations in Diffusion-MRI: Effect of the substrate complexity and parameter choice on the reproducibility of results." pith.science (2026). https://pith.science/paper/4LNC6ZPD

@misc{pith2026190811203,
  author       = {Pith},
  title        = {Pith review of: Robust Monte-Carlo Simulations in Diffusion-MRI: Effect of the substrate complexity and parameter choice on the reproducibility of results},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4LNC6ZPD}},
  note         = {Machine review of arXiv:1908.11203}
}
read the original abstract

Monte-Carlo Diffusion Simulations (MCDS) have been used extensively as a ground truth tool for the validation of microstructure models for Diffusion-Weighted MRI. However, methodological pitfalls in the design of the biomimicking geometrical configurations and the simulation parameters can lead to approximation biases. Such pitfalls affect the reliability of the estimated signal, as well as its validity and reproducibility as ground truth data. In this work, we first present a set of experiments in order to study three critical pitfalls encountered in the design of MCDS in the literature, namely, the number of simulated particles and time steps, simplifications in the intra-axonal substrate representation, and the impact of the substrate's size on the signal stemming from the extra-axonal space. The results obtained show important changes in the simulated signals and the recovered microstructure features when changes in those parameters are introduced. Thereupon, driven by our findings from the first studies, we outline a general framework able to generate complex substrates. We show the framework's capability to overcome the aforementioned simplifications by generating a complex crossing substrate, which preserves the volume in the crossing area and achieves a high packing density. The results presented in this work,along with the simulator developed, pave the way towards more realistic and reproducible Monte-Carlo simulations for Diffusion-Weighted MRI.

Figures

Figures reproduced from arXiv: 1908.11203 by the authors.

Figure 1
Figure 1. Gamma distributed radii and corresponding intra-axonal diffusion signal. Left panel: The distri [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Examples of the curved meshes used as intra-axonal substrates in this study, for three different [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Optimisation procedure of initial trajectories. Left panel: initial trajectories parametrised as a set of [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Top panel shows a visualisation of the resulting fibre crossing substrate after the strand refinement [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: RMAE for each repetition and sampled size for (left) the number of samples and (right) number of [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]
Figure 6
Figure 6. Figure 6: Heat map of the mean RMAE for all the combinations between the number of steps and the number [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]
Figure 7
Figure 7. Figure 7: Tables of the fitting results. Left column shows the fitted intervals of the original ex-vivo ActiveAx [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: The in-between RMAE of the analytical signal of a cylinder, obtained using the GPD approximation, [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: Results for 3 substrates, with 100, 1,000 and 10,000 cylinders respectively. First row: sampled [PITH_FULL_IMAGE:figures/full_fig_p028_9.png]
Figure 10
Figure 10. Figure 10: Mean and standard deviation of the radial DW-MRI signal as a function of substrate size. The [PITH_FULL_IMAGE:figures/full_fig_p029_10.png]
Figure 11
Figure 11. Figure 11: From the leftmost to the right: diffusion tensor map, the resulting fractional anisotropy and [PITH_FULL_IMAGE:figures/full_fig_p029_11.png]
Figure 12
Figure 12. Figure 12: The ICVF maps of one volume slice in the XZ-plane in three different resolutions. The highest [PITH_FULL_IMAGE:figures/full_fig_p030_12.png]
Figure 13
Figure 13. Figure 13: Axon diameter estimation maps (left column) of the regions highlighted in Figure 12, and diameter [PITH_FULL_IMAGE:figures/full_fig_p031_13.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.