---
license: cc-by-4.0
task_categories: [other]
tags: [mri, diffusion-mri, monte-carlo, microstructure, replay-pack, rpk, restricted-diffusion]
pretty_name: Canonical Pores — Monte-Carlo Replay Packs
---

# Canonical Pores — Monte-Carlo Replay Packs

Converged Monte-Carlo diffusion walks in the three geometries that have exact analytical solutions —
**parallel planes, cylinders, spheres** — frozen so that **any** acquisition can be computed afterwards
without re-simulating.

**600 substrates**: 200 diameters per shape, **0.1–20.0 µm in 0.1 µm steps**, each walked to
**T = 200 ms** at D₀ = 2.0×10⁻⁹ m²/s. One `.rpk` (safetensors) per substrate.

## What you can replay

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

| you supply | how it is handled | exact? |
|---|---|---|
| **any gradient waveform** `G(t)` — PGSE, OGSE, PGSTE, CPMG-with-gradients, b-tensor (STE/PTE), arbitrary | projected onto the pack's temporal basis; the gradient phase is *linear* in position, so Parseval applies | exact for waveform content within the stored bands |
| **surface relaxivity** ρ, any value | the pack's boundary-local-time channel, replayed | **exact** — not a mean-field `exp(-TE·ρ·S/V)` tag-on |
| **per-compartment T₁ / T₂** | compartment map + relaxation log-weights | exact |
| **any b-value, Δ, δ, direction, orientation** | consequences of the waveform above | — |

There is **no short-pulse (SGP) and no Gaussian-phase approximation** anywhere in that path. That is the
point of the dataset: the analytical models (`P2`–`P5`, `C2`–`C5`, `S2`–`S5` in
[dmipy-fit](https://github.com/dmrai-lab/dmipy-fit)) each hold under some acquisition assumption; replay
holds for the acquisition you actually ran.

**Not in these packs**: susceptibility / off-resonance (no field channel), magnetization transfer, and
diffusivity is fixed per pack — D₀ is baked into the walk, not a replay knob. (Time-rescaling to a
different D₀ is a planned extension.)

## What has been *certified*, and what that does not mean

Every substrate's replay was compared against the exact eigenmode (matrix-method) solution and sized until
the worst residual met **ε = 5×10⁻³**, measured over a battery of PGSE acquisitions:
(δ, Δ) ∈ {(10,20), (10,40), (10,60), (10,100), (5,40)} ms × b ∈ {1, 3, 5}×10⁹ s/m², plus a
surface-relaxivity battery. Walker counts were **measured**, never extrapolated: each candidate count was
replayed and compared to the analytic reference on up to five *disjoint* blocks, all of which had to pass.

**That battery is where the bound was measured — it is not a limit on what the pack replays.** The
distinction matters in one direction only: a waveform whose temporal content falls outside the tested
spectral range (very high-frequency OGSE, pulses shorter than the save grid) is still *replayable*, but its
error is not covered by the quoted ε. For these three geometries you can settle that yourself — the exact
analytical model ships in dmipy-fit (`P5PlaneMatrixMethod`, `C5CylinderMatrixMethod`,
`S5SphereMatrixMethod`), so any acquisition you care about can be checked directly against ground truth.
That is precisely what makes this dataset the validity proof for replay on substrates where no analytic
answer exists.

Practical resolution limits: the save grid is n_t ∈ {2000, 4000, 8000} over 200 ms, so features finer than
that grid are not resolved; and the position codec keeps K ∈ {128, 196} temporal bands, which bounds the
waveform bandwidth represented.

| shape | substrates | all meet ε=5e-3 | worst residual |
|---|---|---|---|
| plane | 200 | ✅ | see `manifest.json` |
| cylinder | 200 | ✅ | see `manifest.json` |
| sphere | 200 | ✅ | see `manifest.json` |

Each pack carries its own certification in metadata (`grid.worst_residual`, `grid.ladder`), so no consumer
has to trust this table.

## Tiers: fetch only what you need

Two independent selections, and they **compose into a single contiguous byte range** — verified against
HuggingFace range requests (`Accept-Ranges: bytes`, HTTP 206).

**1. Precision (walkers).** Walkers are the leading axis of every per-walker tensor, and any prefix is an
unbiased sub-ensemble with floor ∝ 1/√n. Each pack ships the **measured** prefix length for each floor:

```python
grid["prefix_tiers"]   # e.g. {"0.03": 1000, "0.02": 1000, "0.01": 1000, "0.005": 1000}
```

**2. Spatial axis.** Positions are stored as **one tensor per axis** (`pos_x`, `pos_y`, `pos_z`, each
`(n_walkers, K)`), so you fetch only the components your geometry needs — a slab restricts one direction
(the other two are free and analytic), a cylinder two, a sphere three. All three are always present, so
rotating / b-tensor encodings that need the joint trajectory lose nothing.

Measured on a 17.36 MiB pack: one axis + a 2,000-walker prefix = **0.49 MiB, a 36× reduction**.

```python
from safetensors import safe_open
with safe_open("cylinder/d10.00um.rpk", framework="np") as f:
    x = f.get_slice("pos_x")[0:2000]        # one axis, coarse tier: one contiguous read
```

**Status of the two selections.** Both are real and verified at the byte level, but only one is automatic
today:

* **Per-diameter fetch is automatic.** The `X6` compartments resolve a fitted diameter to the two
  bracketing packs and download only those, so a fit transfers ~tens of MiB, never the dataset.
* **Coarser-ε prefix fetch is automatic too**, via `eps=`:

```python
from dmipy_fit.data.mc_replay import load_replay_family
fam = load_replay_family("cylinder", 2e-9, eps=1e-2)   # range-reads each pack down to its 1e-2 tier
```

  Measured live on this dataset (cylinder d=13.45 µm, 50,000-row pack): ε=5e-3 moves 43.39 MiB,
  ε=1e-2 moves 17.36 MiB, ε=3e-2 moves **4.34 MiB** — a 10× reduction — with the delivered accuracy
  tracking the request. The tier count comes from `manifest.json`, so choosing one costs no transfer,
  and an ε with no measured tier raises rather than silently returning something coarser.

  `axes=` restricts which position components are fetched (1 for a slab, 2 for a cylinder's transverse
  plane) for a further 1.5–3×. It is **not** defaulted: rotating / b-tensor encodings need the joint
  trajectory, so ask for it only when your encoding is fixed-direction.

## Usage

```python
from dmipy_sim import read_rpk, compile_scheme, replay_signal
pack = read_rpk("cylinder/d05.00um.rpk")
W = compile_scheme(G, dt, pack.meta["compression"]["K"])   # G: (n_meas, n_t, 3), any waveform
E = replay_signal(pack, W)                                  # + rho_over_D= for surface relaxivity
```

As fit compartments (`X6`), with the same parameters as their analytic siblings:

```python
from dmipy_fit.signal_models.cylinder_models import C6MonteCarloReplayCylinder
model = C6MonteCarloReplayCylinder()        # mu, lambda_par, diameter — fetches packs on demand
```

`manifest.json` lists every substrate with its walker count, size, certified residual and prefix tiers.

## Layout

```
canonical/D0-2.00e-9/{plane,cylinder,sphere}/d<NN.NN>um.rpk
manifest.json
```

## Citation

Generated with [`dmipy-sim`](https://github.com/dmrai-lab/dmipy-sim). Please cite the dataset DOI and the
dmipy-sim replay-pack reference. CC-BY-4.0.