---
license: cc-by-4.0
task_categories: [other]
tags: [mri, diffusion-mri, monte-carlo, microstructure, replay-pack, rpk, myelin, susceptibility, axon]
pretty_name: Winther G6 Axons — Monte-Carlo Replay Packs with Susceptibility
---

# Winther G6 Axons — Monte-Carlo Replay Packs

Twenty-nine **real myelinated axons**, segmented from synchrotron X-ray nano-holotomography of monkey
corpus callosum, each walked once by a Monte-Carlo diffusion simulator and frozen so that **any**
acquisition can be computed afterwards without re-simulating.

**The morphology is not ours.** It is the `G6` configuration set of Winther et al. (2024), distributed by
them under CC-BY-4.0. This dataset contributes the computation and a documented, self-certifying container.
Meshes are used as published — full length, no cropping, smoothing or re-meshing.

![axon cross-sections](figures/fig_substrate.png)

*Cross-sections of `axon06` along its length, plus a 3-D view. Axon (inner) and myelin (outer) surfaces as
published — the calibre and tortuosity vary along the fibre, which is the morphology under study.
Generated by `report_assets.fig_substrate`.*

## What you can replay

A replay pack stores the walk, not a table of pre-computed signals, so the forward model is evaluated at
read time:

| you supply | exact? |
|---|---|
| any gradient waveform (PGSE, PGSTE, OGSE, CPMG, free-form) | yes, within the stored temporal band |
| B₀ magnitude and direction | yes — the susceptibility basis is geometry-only |
| isotropic **and** anisotropic myelin susceptibility | yes |
| surface relaxivity ρ | yes — boundary local time is stored (C2) |
| bulk T₂ / T₁ per compartment | yes (C1) |

## Contents

```
packs/axonNN.rpk         29 replay packs   (~71 MB each)
field/axonNN.field.rpk   29 susceptibility field companions, masked (14–36 MB each)
manifest.json            per-substrate metadata + SHA-256
figures/                 the figures on this page, and the scripts that make them
```

Total 2.54 GiB. The field companions are stored **masked** — an axon meanders, so a bounding box around it
is ~92% empty, and only the voxels a walker can reach are kept (10.1× smaller, values identical at the
stored voxels). The mask is derived from the companion itself, so no mesh is needed to use one.

## Provenance and fidelity

| | |
|---|---|
| diffusivity D₀ | 0.6 × 10⁻⁹ m²/s (ex vivo, source study) |
| echo time | 36 ms |
| myelin χ_iso | +1.06 × 10⁻⁶ (source study); Δχ_a = 0 as the reference value |
| field grid | 0.131 µm, partial-volume myelin mask, no k-space apodisation |
| walkers seeded | 52,000 per axon at uniform density (intra + frozen myelin) |
| codec error | 6.96e-04 – 1.09e-03 (median 8.76e-04) |
| Monte-Carlo floor | 4.88e-03 – 1.98e-02 (median 9.33e-03) |
| self-certifying | **29 / 29** (codec error below the pack's own floor) |

Each pack **measures and stores its own replay fidelity** against its Monte-Carlo floor, so the error you
would incur is a property you can read, not one you have to trust.

Quote the **per-axon** floor rather than a dataset-wide number: it spans 4× across the set because it
tracks each substrate's own internal-gradient variance, not the compression.

## Is the physics right?

Against an **exact** answer, not against another simulation. For an infinite coaxial hollow cylinder the
answer is known in closed form, with no fitting freedom to hide an error in:

* the field inside the lumen is **exactly zero** at every orientation;
* inside the sheath at θ = 90°, `ΔB/B₀ = χ[−1/6 − ½(R_i²/r²)cos 2φ]`;
* inside a *solid* cylinder the interior is uniform at `χ/6·(3cos²θ − 1)`.

![analytic validation](figures/fig_analytic_validation.png)

| | measured | exact |
|---|---|---|
| lumen field | **0.131%** of χ·B₀ | 0 |
| sheath amplitude | **0.9996×** analytic | 1 |
| sheath structure, signed | **corr +0.9995, slope +0.999** | +1 |
| solid-cylinder interior (θ=90°) | **−0.1649** χ·B₀ | −1/6 |

All are permanent gates in the generator's test suite (`test_susceptibility_field_oracle.py`), not one-off
checks, and the sheath comparison is **signed** — an inverted field fails it. The lumen null is the sharp
one: a hard binary myelin source rings into the lumen at ~2.6% of χ·B₀ through the non-decaying dipole
kernel, which would silently inflate intra-axonal dephasing; partial-volume occupancy suppresses it to
~0.1%.

## Susceptibility dephasing vs B₀ orientation

![B0 rotation](figures/fig_b0_rotation.png)

Intra-axonal spin-echo signal at b = 0 as B₀ rotates from parallel to the fibre (0°) to perpendicular
(90°) — no diffusion weighting, so this isolates the susceptibility dephasing. Every one of the 29 axons
is shown.

The angular dependence matches the source study in shape (both minimise at 90°). The **magnitude differs**:
contrast 0.037 here against 0.015 published, i.e. ~2.4× more attenuation at 90°. That comparison, and the
field-solver question it raises, is analysed separately rather than resolved on this page — note only that
the two are validated against different references, and the closed-form check above is the one with an
exact answer.

## Reproduce the curve yourself

The figure above is not stored — it is **computed from one pack at read time**. This is the whole point of
the format, so here it is in full:

```python
import numpy as np
from huggingface_hub import hf_hub_download
from dmipy_sim import bank
from dmipy_sim.bank import read_rpk

path = hf_hub_download("SubstrateCommons/winther-g6-axons", "packs/axon06.rpk",
                       repo_type="dataset")
pk = read_rpk(path)
TE = (pk.n_t - 1) * pk.dt

class B0Only:                       # b = 0: isolate the susceptibility dephasing
    G = np.zeros((1, pk.n_t, 3))
    dt = pk.dt

signal = []
for deg in (0, 15, 30, 45, 60, 75, 90):
    t = np.deg2rad(deg)
    s = bank.replay_susc(pk, B0Only, b0_dir=[np.sin(t), 0.0, np.cos(t)],
                         B0=7.0, chi_iso=1.06e-6, refocus_time=TE / 2,
                         relaxation=False, complex_signal=True, compartment=1)
    signal.append(float(np.real(s[0])))

print([round(x, 4) for x in signal])
# [0.9243, 0.9337, 0.9457, 0.9462, 0.9322, 0.9159, 0.9121]
```

Nothing in that loop was decided when the pack was built. `B0=7.0` could be 3, `chi_iso` could be anything
including an anisotropic component, `compartment=1` could be the myelin pool, `B0Only.G` could be any
gradient waveform you like, and `refocus_time` places the spin echo wherever you want it. Each is a
parameter of the **replay**, not of the simulation — which is why one 71 MB file answers a question nobody
asked when it was written. Requires `pip install dmipy-sim`.

## Seeding and confinement

The intra-axonal pool is seeded by an **exact ray-parity containment test**, and confinement is verified
against independent ray parity rather than against the seeding test itself: of 509 genuinely interior
seeds, 1.4% lie outside the surface at TE, and those end 0.07 µm beyond the wall — within the
one-triangle accuracy of the test that measures them.

## Citation

The substrate morphology is the source study's; please cite it:

> Winther et al., *Susceptibility-induced internal gradients reveal axon morphology and cause anisotropic
> effects in the diffusion-weighted MRI signal*, Sci. Rep. **14**:29636 (2024).
> doi:10.1038/s41598-024-79043-5 — morphology dataset resources.drcmr.dk, CC-BY-4.0

Packs generated with [`dmipy-sim`](https://github.com/dmrai-lab/dmipy-sim).