Results

χ-separation (2026)

Susceptibility source separation splits the net susceptibility χ into a paramagnetic part χ⁺ (iron) and a diamagnetic part χ⁻ (myelin, calcium). Plain QSM measures only their signed sum, χ_total = χ⁺ + χ⁻, in which the two partly cancel, so it cannot tell them apart. The reversible relaxation rate R2′ responds instead to their combined magnitude, |χ⁺| + |χ⁻|. Having both the signed sum and the magnitude gives two independent constraints per voxel, which is what makes the sources recoverable. Each method is given χ_total, R2′, the local field and the multi-echo signals, and must produce χ⁺ and χ⁻, which are scored separately.

The phantoms

Four phantoms are scored independently, all built on the Challenge 2.0 head model (Marques et al., 2021) via qsm-forward, and switchable with the leaderboard’s phantom selector. The first three reproduce the Susceptibility-Separation-Phantom of Ridani et al. (2026), verified voxelwise against the authors’ own ground truth (isotropic χ⁻ matches to < 10⁻⁵ ppm; per-ROI means to ≤ 5×10⁻⁴ ppm); the fourth, the Stewart multicompartment phantom (2026), is a QSM-CI extension that makes white-matter anisotropy recoverable from the signal itself.

The three Ridani sets are noiseless and gradient-echo-only, with a mono-exponential white-matter decay in which the fibre angle θ modulates only the scalar R2′. They deliberately omit a fibre-orientation input so orientation cannot be handed to methods — a closed-form solver given the fibre angle separates the anisotropic phantom almost exactly.

The Stewart multicompartment phantom adds a matched spin-echo and replaces that mono-exponential decay with a three-pool hollow-cylinder white-matter signal model (Wharton & Bowtell, 2012). Its orientation-dependent pool interference produces a non-mono-exponential GRE decay whose beat encodes the fibre angle, so θ can be recovered from the signal itself rather than needing DTI. It is the minimal model in which fibre angle leaves any mark on the signal: mono-exponential decay carries none, and fuller models (fibre dispersion, exchange, diffusion) add realism at the cost of recoverability — the hollow-cylinder sits at the point where anisotropy is realistic enough to matter yet still cleanly recoverable.

Across all four, χ⁺ and χ⁻ are built per tissue and summed to give χ_total, so the net susceptibility shares the same anatomy as the In silico dataset. All are generated at 1 mm, chosen so that every method — including the memory-hungry deep-learning ones — runs within standard CI limits.

The forward model

Building χ⁺ and χ⁻

Each tissue is assigned literature χ⁺ and χ⁻ values (χ⁺_lit, χ⁻_lit), with within-tissue texture from the acquired R1 and R2* maps:

where the bar denotes a tissue-mean (\(\overline{R_2^*}\) and \(\overline{R_1}\) are the average R2* and R1 over that tissue, so each texture term is this voxel’s deviation from the tissue average), and the modulation coefficients a and b are set per tissue and per source. Summing the two gives χ_total.

χ_total is one of the two constraints from above; the other is R2′, which relates to the source magnitudes through a relaxivity D_r:

If D_r were a single known constant, the two would solve directly, a closed-form null solution:

This null solution is a useful baseline: it adds nothing beyond the standard assumption, so any method should at least match it. Real separation is harder because the true relaxivity is not a single known constant. Its main complication, described next, is that the myelin relaxivity depends on fibre orientation, which the field and R2′ alone cannot reveal; it also differs between iron and myelin, but that is a smaller, calibratable offset. That orientation dependence is why χ⁻ is the harder map to recover, and it is what the standard separation model (which assumes isotropic sources) leaves out.

Methods improve on this baseline by relaxing one of its assumptions: estimating the relaxivity rather than fixing it to a single constant, fitting the multi-echo signal decay directly instead of relying on the scalar R2′, or imposing spatial and learned priors that the voxel-wise closed form has none of.

White-matter anisotropy

Myelin is magnetically anisotropic, so in white matter χ⁻ depends on the angle θ between the nerve fibre and B₀, taken from the diffusion V1 eigenvector:

where Δχ is the myelin susceptibility anisotropy (susceptibility along the fibre minus across it) and χ₀ the orientation-independent part, both per tract. A single-orientation acquisition cannot observe θ, so the myelin relaxivity D_r⁻(θ) is unknown per voxel, which is what breaks the null solution for χ⁻. Iron is modelled as isotropic spheres and carries no such term.

R2′ and the relaxivity D_r

The true relaxivity is source-dependent, so R2′ is built as

D_r comes from the static-dephasing model (Yablonskiy & Haacke, 1994): spheres for χ⁺, cylinders for χ⁻, giving D_r⁺ = 107.84·B₀ for iron and D_r⁻ = 133.77·sin²θ·B₀ for myelin (755 and 936·sin²θ Hz/ppm at 7 T). These grow with field and run about 5× higher than measured in vivo — not an error, but the idealisation the model makes: static dephasing assumes perfectly static spins around ideal spheres/cylinders, whereas real tissue has diffusion averaging and heterogeneous microstructure that lower the effective relaxivity, so empirical calibrations (e.g. Shin et al.) land well below the theoretical value.

The three Ridani phantoms use the scaled setting — the static-dephasing values above, field-consistent and matching the reference exactly. The multicompartment phantom uses a fixed setting: both relaxivities are rescaled by a single factor so the iron value is D_r⁺ = 320 Hz/ppm (the field-scaled empirical calibration of Shin et al., 2021 at 7 T), field-independent, with the same χ⁺/χ⁻ ground truth.

Scaled vs fixed. Scaled is field-physical and reproduces Ridani; on such a phantom, methods that assume a lower empirical D_r carry a scale offset (an informative, real model error). Fixed anchors D_r⁺ to the empirical value so the score reflects separation rather than D_r calibration.

The signal

The simulated multi-echo GRE magnitude carries the separated sources through R2′ rather than a single lumped R2*, so its decay is written directly in terms of χ⁺ and χ⁻:

so the effective decay rate is R2* = R2 + R2′. On the multicompartment phantom the matched spin-echo removes the reversible term and decays with R2 alone (\(S(\mathrm{TE}) = M_0\,e^{-\mathrm{TE}\,R_2}\)), letting a signal-domain method estimate R2′ = R2* − R2 itself instead of relying on the provided R2′ map.

The multicompartment signal (χ-sep MC)

On the Ridani phantoms the fibre angle θ enters only as a scalar that modulates the white-matter R2′ — so θ is not recoverable from a single-orientation acquisition, and white-matter χ⁻ is unbeatable without an external DTI input. The multicompartment phantom removes that ceiling: white matter is modelled as three water pools (myelin, axonal, extra-axonal) with orientation-dependent frequency offsets (a hollow-cylinder model, Wharton & Bowtell, 2012). The pools beat against each other, giving a non-mono-exponential magnitude whose shape encodes θ and the myelin-water fraction — so a method can recover both from the GRE signal alone. This makes it the only phantom where anisotropy-aware separation can win without being handed the fibre orientation.

Scoring

The headline metric is the per-ROI MSPE (mean squared percentage error of ROI means) of Ridani et al. (2026), so the numbers are directly comparable to the paper: χ⁺ is averaged over the deep-gray-matter iron nuclei and cortical gray matter, χ⁻ over the white-matter fibre bundles of a co-registered atlas. Detrended NRMSE (scale-independent, voxelwise) and xSIM are also reported per source. For the anisotropic phantoms a MEV diagnostic (maximum error variation) reports how much the χ⁻ error falls from fibres parallel to B₀ to perpendicular. Each phantom also carries a null baseline — the closed-form solve with the phantom’s own assumptions — that any real method must beat.

Differences from the reference phantom

The three Ridani phantoms reproduce the χ⁺/χ⁻ construction, white-matter anisotropy and acquisition of Ridani et al. (2026), and their ground truth is verified voxelwise against the authors’ own OSF data. A few implementation choices differ:

• Orientation withheld. No fibre-angle (DTI) input is shipped, so a method cannot be handed the orientation that would make the anisotropic phantom nearly closed-form solvable. It must recover white-matter χ⁻ from the field and R2′ alone (or, on the multicompartment phantom, from the signal beat).
• Seeded noise. The reference’s white-matter texture noise is seeded here for reproducibility, so the ground truth is bit-reproducible; the authors’ unseeded realisation is matched only in distribution.
• Sign enforced. χ⁺ is clamped ≥ 0 and χ⁻ ≤ 0 (a source map cannot flip sign).
• Multicompartment (χ-sep MC only). The 7 T fixed-D_r phantom additionally simulates the hollow-cylinder white-matter signal and a matched spin-echo — a QSM-CI extension beyond the reference’s mono-exponential model.

The diffusion V1 eigenvector is stored mirrored (L-A-S) relative to the segmentation; qsm-forward flips it to register the fibres, which reproduces the reference’s published fibre-angle map exactly. The public reference code omits this flip.

Reproduce the phantoms

qsm-forward builds the phantoms from a data directory holding the Challenge 2.0 tissue-property maps and diffusion V1 eigenvector. Request those from the Donders repository (data.ru.nl/…/DSC_3015069.02_542, DOI 10.34973/m20r-jt17), and add the white-matter tract mask from the Susceptibility-Separation-Phantom repository (Ridani et al.). QSM-CI’s scripts/gen_chisep.py then generates each phantom from a preset:

gen_chisep.py --preset ridani-3t-iso    --out data/ridani-3t-iso
gen_chisep.py --preset ridani-3t-aniso  --out data/ridani-3t-aniso
gen_chisep.py --preset ridani-7t-aniso  --out data/ridani-7t-aniso
gen_chisep.py --multicompartment --dr-model fixed --dr-fixed 320 \
    --tes 3,9,15,21,27,33,39,45 --out data/chisep-mc   # χ-sep MC

Each preset sets the field, echo train, isotropy and relaxivity, simulates the signals and flattens the result into the dataset layout (public inputs/, held-out groundtruth/); point qsm-ci run straight at the NIfTIs.

Citations

• Separation phantom: Ridani D., De Leener B., Alonso-Ortiz E. “A realistic in-silico brain phantom for quantifying susceptibility anisotropy-induced error in susceptibility separation.” Magn Reson Med 2026. doi:10.1002/mrm.70468.
• Head model: Marques J.P., Meineke J., Milovic C., et al. “QSM reconstruction challenge 2.0: a realistic in silico head phantom for MRI data simulation and evaluation of susceptibility mapping procedures.” Magn Reson Med 2021;86(1):526–542. doi:10.1002/mrm.28716.
• χ-separation model & D_r: Shin H.G., Lee J., Yun Y.H., et al. “χ-separation: Magnetic susceptibility source separation toward iron and myelin mapping in the brain.” NeuroImage 2021;240:118371. doi:10.1016/j.neuroimage.2021.118371.
• Static-dephasing regime: Yablonskiy D.A., Haacke E.M. “Theory of NMR signal behavior in magnetically inhomogeneous tissues: the static dephasing regime.” Magn Reson Med 1994;32(6):749–763. doi:10.1002/mrm.1910320610.
• DECOMPOSE: Chen J., et al. “Decompose QSM to sub-voxel diamagnetic and paramagnetic components based on gradient-echo MRI data.” NeuroImage 2021. doi:10.1016/j.neuroimage.2021.118735.
• SUSEP-Net: Li, Gao, Sun, et al. “SUSEP-Net: simulation-supervised contrastive learning for susceptibility source separation via a sub-voxel QSM network.” arXiv:2506.13293, 2025. doi:10.48550/arXiv.2506.13293.
• WaveSep: Fang Z., Shin H.G., van Zijl P., Li X., Sulam J. “WaveSep: A Flexible Wavelet-Based Approach for Source Separation in Susceptibility Imaging.” MLCN, MICCAI 2023. doi:10.1007/978-3-031-44858-4_6.
• R2*-QSM: Dimov A.V., et al. “Magnetic Susceptibility Source Separation Solely from Gradient Echo Data: Histological Validation.” Tomography / J Neuroimaging 2022. doi:10.1111/jon.13014.
• Multicompartment white-matter signal: Wharton S., Bowtell R. “Fiber orientation-dependent white matter contrast in gradient echo MRI.” PNAS 2012;109(45):18559–18564. doi:10.1073/pnas.1211075109.

In vivo · 2016 Reconstruction Challenge

This dataset is real acquired brain data from the 2016 QSM Reconstruction Challenge: a single-orientation 3 T scan of one volunteer. Each method inverts the ground-truth local tissue field to a susceptibility map, scored against two reference reconstructions rather than a true susceptibility (none exists in vivo). Only the dipole-inversion stage is scored, so the ranking reads as agreement with a reference, not accuracy against ground truth.

The challenge and its references

The 2016 Reconstruction Challenge (Langkammer et al., 2018) took the χ₃₃ component of a susceptibility-tensor (STI) reconstruction as its reference, hoping to sidestep the orientation bias of COSMOS. Twenty-seven submissions were scored on RMSE, HFEN, SSIM and regional error. The lesson was as much about metrics as algorithms: tuned to minimise those errors, the winning maps were over-smoothed and lost fine structure, and near-identical scores hid visibly different maps. A follow-up (Milovic et al., 2020) then showed the χ₃₃ reference itself did not match the measured phase once anisotropy, microstructure and noise were accounted for.

QSM-CI scores each method against both available references: the multi-orientation COSMOS reconstruction and the STI χ₃₃ map. Source: the 2016 challenge (qsm.neuroimaging.at); the local tissue field and both reference χ maps are in ppm.

Citations

• 2016 Challenge report: Langkammer C., Schweser F., Shmueli K., et al. “Quantitative susceptibility mapping: Report from the 2016 reconstruction challenge.” Magn Reson Med 2018;79(3):1661–1673. doi:10.1002/mrm.26830.
• 2016 Challenge lessons: Milovic C., Tejos C., Acosta-Cabronero J., et al. “The 2016 QSM Challenge: Lessons learned and considerations for a future challenge design.” Magn Reson Med 2020;84(3):1624–1637. doi:10.1002/mrm.28185.
• COSMOS reference: Liu T., Spincemaille P., de Rochefort L., Kressler B., Wang Y. “Calculation of susceptibility through multiple orientation sampling (COSMOS).” Magn Reson Med 2009;61(1):196–204. doi:10.1002/mrm.21828.