Trusting the Inverse: Reliability-Aware Mapping for Simulation-Based Microstructure Estimation in Diffusion MRI

Introduction

• DW-MRI measures tissue microstructure non-invasively, and white-matter models use its signals to estimate properties including axon diameter, neurite density, and orientation dispersion.These estimates are investigated as candidate biomarkers for neurodegeneration, demyelination, and abnormal development, motivating assessment of how strongly signals constrain tissue configuration.Source: Introduction · p. 1
• White matter includes broad axon-diameter distributions, non-cylindrical geometries, and substantial morphological variability along axons.These features make white matter more complex than the assumptions underlying most signal models.Source: Introduction · p. 1
• Monte Carlo diffusion simulations generate synthetic signals within three-dimensional substrates that can represent irregular axonal cross-sections, heterogeneous packing, and realistic spatial organisation.Such signals can be used to construct dictionaries for simulation-based estimation.Source: Introduction · pp. 1–2
• Distinct tissue configurations can produce nearly identical diffusion signals, leaving multiple plausible parameter solutions.Existing uncertainty methods generally do not identify whether unreliability comes from out-of-distribution signals, insufficient local signal representation, or parameter degeneracy.Source: Introduction · p. 2
• The paper introduces three complementary reliability scores for out-of-distribution signals, local signal mismatch, and parameter degeneracy.The scores characterize reliability relative to the simulated dictionary and acquisition protocol.Source: Introduction · p. 2
• The framework uses a 1,050-voxel Monte Carlo dictionary built from CACTUS substrates and MC-DC simulations, with parameter ranges grounded in rat corpus callosum electron microscopy.The authors apply it to in vivo rat and human corpus callosum DW-MRI.Source: Introduction · p. 2

In vivo rat corpus callosum DW-MRI

• The rat analysis used 256 manually segmented corpus callosum voxels from four animals, without an FA threshold, making it a proof-of-concept rather than a powered population analysis.The longest available diffusion time was selected for each animal to maximise sensitivity to microstructural restriction.Source: In vivo rat corpus callosum DW-MRI · pp. 2–3

In vivo human corpus callosum DW-MRI

• The human analysis included 18,765 corpus callosum voxels from nine of 35 MGH-USC HCP subjects, acquired at 3 T and 1.5 mm isotropic resolution.Spherical-mean signals at b-values of 1,000, 3,000, 5,000, and 10,000 s/mm2 were matched to the dictionary.Source: In vivo human corpus callosum DW-MRI · p. 3
• CACTUS substrates used parameter ranges grounded in rat corpus-callosum histology; 175 combinations of radius, microscopic angular spread, and packing density, paired with six intrinsic-diffusivity values, produced a 1,050-voxel dictionary.Radii ranged from 0.25 to 0.85 µm in 0.1 µm increments; angular spread ranged from 0° to 10° in 2.5° steps; packing density ranged from 60% to 92% in 8% increments.Source: Histology-informed substrate… · p. 3
• MC-DC simulations represented intra- and extra-axonal contributions, excluded myelin signal, and replicated each dataset’s acquisition protocol for measured–simulated comparisons.The compartment weight is geometric ICVF because compartment-specific T2 relaxation was not imposed and the common factor cancels in b0 normalisation.Source: Diffusion MRI signals via Monte Carlo… · p. 3
• Voxel estimates used rotationally invariant spherical means, log-MAE distances, and exponentially weighted KNN regression with K = 10 neighbours and α = 10.Marginal 95% confidence intervals were estimated by 500 bootstrap resamples of the neighbour set.Source: Parameter estimation · p. 4

Three reliability scores: Outlier-Detection, Signal-Matching, and Parameter-Degeneracy

• Three scores target out-of-dictionary extrapolation, poor local signal reconstruction, and parameter degeneracy, then combine geometrically into a Reliability Index.The scores use local outlier factor, normalized reconstruction residual, and normalized neighbour-parameter spread, respectively.Source: Three reliability scores:… · pp. 4–5, Table 1 · p. 6
• The Reliability Index is the geometric mean of the three scores, with reliable estimates defined as R > 0.60, moderate as 0.40–0.60, and unreliable as R < 0.40.Per-parameter precision reports the fraction of each plausible parameter range outside its ±1σ neighbour interval.Source: Three reliability scores:… · p. 6

Self-validation of the Reliability Index on the synthetic dictionary

• Each of 1,050 synthetic voxels was tested against the other 1,049 dictionary entries after Rician noise was added at nine SNR levels, yielding 9,450 cases.Noisy signals were estimated and scored with exponentially weighted KNN (K = 10, α = 10) against the clean dictionary.Source: Self-validation of the Reliability… · p. 7
• Predictive value was assessed using Spearman rank correlation between reliability scores and mean relative parameter error across test cases.Low-reliability cases were grouped by the dominant source, defined as the minimum of Sout, Smatch, and Sdeg.Source: Self-validation of the Reliability… · p. 7

Complement-mask analysis

• The complement-mask analysis compared estimation inside the corpus callosum mask with a surrounding region containing tissue types and partial-volume configurations absent from the dictionary.The complement comprised voxels within a bounding box expanded by two voxels around the mask, excluding the mask itself.Source: Complement-mask analysis · p. 7

Self-validation of the Reliability Index across SNR levels

• As noise increased, reliability scores declined; ICVF error stayed below 10% at SNR = 25, while microscopic angular spread had approximately 22% mean relative error across SNR levels.At SNR = 25, radius and intrinsic diffusivity errors reached approximately 30% and 25%, respectively; Sdeg variability increased at higher SNR than for Sout and Smatch.Source: Self-validation of the Reliability… · p. 7
• The Reliability Index correlated negatively with mean relative parameter error across 9,450 synthetic test cases (Spearman ρ = −0.742, p < 10−10).The test used leave-one-out validation across 1,050 synthetic voxels and nine SNR levels.Source: Self-validation of the Reliability… · p. 7

Reliability Index and parameter estimates in rat corpus callosum in vivo

• 91% of rat corpus callosum voxels exceeded R > 0.5; Sout medians were 87–90%, while Sdeg was the lowest score in all four animals.Smatch medians were 94–99% in Rats 1–3 and 68% in Rat 4; Sdeg medians were 31–41% in Rats 1–3 and 52% in Rat 4.Source: Reliability Index and parameter… · p. 8
• Rat corpus callosum median axon radii ranged from 0.34 to 0.57 µm, and estimated ICVF values ranged from 0.60 to 0.74.Microscopic angular spread ranged from 3.3° to 5.3°, and intrinsic diffusivities ranged from 2.08 to 2.70 µm²/ms.Source: Reliability Index and parameter… · p. 8
• Rat 3 had larger radius and ICVF estimates; within rat corpus callosum, reliability was higher centrally, while complement voxels had lower reliability and Sout and estimates near dictionary bounds.Across animals, Sout remained high throughout most corpus callosum voxels; complement estimates frequently approached training-dictionary bounds, with elevated radius and diffusivity and edge-range ICVF.Source: Reliability Index and parameter… · pp. 9–11

Voxel-level examples on in vivo data

• The voxel examples distinguish well-matched signals from poor local interpolation and emerging out-of-support signals, showing that high Sdeg can coexist with low overall reliability.The low-Smatch example had Sout = 69%, Smatch = 4%, and Sdeg = 67%; the complement example had Sout = 56%, Smatch = 28%, and R = 39%.Source: Figure 6 · p. 13, Figure 7 · p. 14, Voxel-level examples on in vivo data · p. 15

Reliability Index in human corpus callosum

• 73% of human corpus callosum voxels exceeded R > 0.5, with per-subject values of 55–77%; reliability was concentrated centrally, and Sdeg was usually the lowest score.Subject 3 was limited by Smatch, with a median of 42%.Source: Reliability Index in human corpus… · p. 15

• ρ = −0.742 (p < 10^-10): leave-one-out validation found that the Reliability Index tracked estimation error across noise levels.Sdeg showed greater variability at higher SNR than Sout and Smatch, indicating that degeneracy can emerge while signal matching and dictionary support remain relatively stable.Source: Discussion and Conclusions · p. 20
• 91% of rat corpus-callosum voxels exceeded R > 0.5, although Sdeg was consistently lower than Sout and Smatch across animals.Because the Reliability Index is a geometric mean, lower Sdeg tempers the combined score rather than rejecting the voxel outright.Source: Discussion and Conclusions · p. 20
• The three scores distinguish dictionary coverage, local reconstruction fidelity, and parameter degeneracy, while covariance projections indicate which parameter combinations contribute to ambiguity.Reading Sout and Smatch together separates a poor local match from a signal leaving the dictionary; the decomposition also helped guide iterative dictionary refinement.Source: Discussion and Conclusions · p. 21
• Human results showed the same pattern, with Sdeg usually the lowest score; Subject 3 was limited by Smatch instead.Examples included R = 97% when all three pillars were strong and R = 4% for a voxel outside dictionary support; the rat-derived dictionary transferred without a collapse in Sout.Source: Discussion and Conclusions · p. 21
• The dictionary represents highly aligned corpus-callosum white matter; whole-brain use would require broader ranges and configurations such as crossing fibres and grey matter.Powder-averaging erodes fibre-orientation information, and simulations also omit compartment-specific T2 relaxation, myelin water, and CSF contributions.Source: Discussion and Conclusions · pp. 21–22
• Without same-animal histology, comparison with prior studies supports biological plausibility rather than direct validation of the in-vivo estimates.The study is a proof of concept based on 256 rat voxels from four animals and 18,765 human voxels from nine subjects.Source: Discussion and Conclusions · pp. 20, 22

Synthetic examples

• R = 93% characterises the high-reliability example, with closely overlapping neighbour curves, tight covariance ellipses, and a well-constrained estimate.Source: Figure S1 · p. 22
• Sout = 35% is the dominant limitation in the low-Sout example, where the noisy signal lies near the dictionary boundary and neighbour curves diverge from the target.The divergence is most pronounced at high b-values, where noise is most pronounced.Source: Figure S2 · p. 23
• Smatch = 32% dominates the low-Smatch example: despite dictionary support and a well-constrained estimate, systematic residuals indicate insufficient local dictionary density for accurate interpolation.Source: Figure S3 · p. 24
• Sdeg = 12% is the dominant limitation in the low-Sdeg example, where overlapping neighbour curves but elongated, spread covariance ellipses indicate parameter degeneracy.Distinct parameter configurations produce nearly identical signals.Source: Figure S4 · p. 25

In vivo supplementary results

• Rat 2 has 94% of voxels above R > 0.5, while Rat 4 has 74% and the lowest reliability, with a lower median Smatch than the other animals.Per-voxel estimates and reliability scores are shown in reliability-layered histograms and per-animal medians in Table 2.Source: Figure S5 · p. 25, Figure S6 · p. 26, Supplementary Material · p. 31
• Supplementary panels provide spatial parameter maps for Rat 2 and Rat 4, plus complement maps for Rats 2, 3, and 4.Source: Figure S7 · p. 27, Figure S8 · p. 28, Figure S9 · p. 29

Funding

• The Swiss National Science Foundation supported the work under grant 205320_204097.Source: Funding · p. 34
• Open access was funded by the École Polytechnique Fédérale de Lausanne.Source: Funding · p. 34

Author contributions statement

• J.L.V.H. co-designed the study, generated substrates and simulations, implemented the estimation and reliability pipeline, performed analyses, and drafted the manuscript.Source: Author contributions statement · p. 34
• J.R.-P. co-designed the study, contributed to methodological development and scientific discussion, supervised the work, and drafted and revised the manuscript.Source: Author contributions statement · p. 34
• I.J. shared the in vivo NEXI dataset and contributed to interpretation and revision; J.-P.T. supervised the project and contributed to interpretation and revision.All authors reviewed and approved the final manuscript.Source: Author contributions statement · p. 34

Data availability

• The in vivo DW-MRI data acquired as part of the NEXI study are available from the corresponding author upon reasonable request.Source: Data availability · p. 34
• The CACTUS framework and MC-DC simulator are publicly available.Source: Data availability · p. 34
• The signal dictionary will be made publicly available on publication.Source: Data availability · p. 34