L1 ⚛ L1-503 ⊙ Testnet

Quantitative Susceptibility Mapping (QSM)

Medical Imaging · Quantitative MR susceptibility recovery (multi-physics joint inverse)

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⬇ L1-503.md

# ⚛ L1 Principle — Quantitative Susceptibility Mapping (QSM) **ID:** `L1-503` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Quantitative MR susceptibility recovery (multi-physics joint inverse)* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D quantitative susceptibility > **📡 Carrier:** radio_wave · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553359 --- ## 🧠 1. Introduction **Quantitative Susceptibility Mapping (QSM)** is a **linear inverse problem** whose unknown lives in **3D quantitative susceptibility** space, within the **Quantitative MR susceptibility recovery (multi-physics joint inverse)** sub-domain of **Medical Imaging**. Measurements consist of radio-frequency electromagnetic waves via a **mri gre phase with maxwell magnetostatic** sensing mechanism. The forward operator applies, in order: Bloch-equation tip of the magnetisation vector; L · gradient echo operator; L · larmor precession operator; L · maxwell magnetostatic operator; L · dipole convolution operator; L · phase extract operator; pixel-level spatial averaging on the detector. Observations are corrupted by additive Gaussian noise. Existence of recovered 3D quantitative susceptibility chi(r) is guaranteed within the declared Omega bounds. Uniqueness holds on the wave-cone-supported subspace; modes inside the magic-angle zero cone are recovered only through priors (MEDI L1-edge, TGV second-order, deep-prior). Stability is moderately conditioned (kappa_eff ~ 30 after wave-cone regularization) — B0_inhomogeneity and background_field_removal_error dominate the stability cliff; phase_unwrap_error and dipole_kernel_truncation contribute higher-order bias terms. Joint Hadamard well-posedness for the coupled phase-field-susceptibility forward is established by Liu, de Rochefort, Wharton, Bilgic literature 2009-2018 (see references). ## ⚙ 2. Forward Model Physical chain: **x** → RF excitation pulse → L · gradient echo → L · larmor precession → L · maxwell magnetostatic → L · dipole convolution → L · phase extract → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.phase_extract` `L.dipole_convolution` `L.maxwell_magnetostatic` `L.larmor_precession` `L.gradient_echo` B₁(t) x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.rf_excitation` | Bloch-equation tip of the magnetisation vector | | `L.gradient_echo` | L · gradient echo operator | | `L.larmor_precession` | L · larmor precession operator | | `L.maxwell_magnetostatic` | L · maxwell magnetostatic operator | | `L.dipole_convolution` | L · dipole convolution operator | | `L.phase_extract` | L · phase extract operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | Quantitative MR susceptibility recovery (multi-physics joint inverse) | | Carrier | radio_wave | | Problem class | linear_inverse | | Solution space | 3D_quantitative_susceptibility | | Noise model | gaussian | | Integration axis | spatial | | Difficulty delta | 5 | | L dag | 7.4 | ## 📡 4. Measurement Model Existence of recovered 3D quantitative susceptibility chi(r) is guaranteed within the declared Omega bounds. Uniqueness holds on the wave-cone-supported subspace; modes inside the magic-angle zero cone are recovered only through priors (MEDI L1-edge, TGV second-order, deep-prior). Stability is moderately conditioned (kappa_eff ~ 30 after wave-cone regularization) — B0_inhomogeneity and background_field_removal_error dominate the stability cliff; phase_unwrap_error and dipole_kernel_truncation contribute higher-order bias terms. Joint Hadamard well-posedness for the coupled phase-field-susceptibility forward is established by Liu, de Rochefort, Wharton, Bilgic literature 2009-2018 (see references). | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | RMSE_ppm | ## 📏 5. Operating Range (Ω) **Center problem class:** `qsm` · **Forward operator:** `qsm_joint_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 256 | | W | px | 256 | | Z | — | 64 | | Snr db | dB | 24 | | N echoes | — | 6 | | Te ms max | — | 30 | | Te ms min | — | 5 | | B0 field t | — | 3 | | Voxel size mm | mm | 1 | | B0 inhomogeneity | — | 0 | | Phase unwrap error | — | 0 | | Brain segmentation error | — | 0 | | Background field removal error | — | 0 | | Dipole kernel truncation error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 128 – 512 | | W | px | 128 – 512 | | Z | — | 32 – 256 | | Snr db | dB | 10.0 – 40.0 | | N echoes | — | 1 – 12 | | Te ms max | — | 10.0 – 80.0 | | Te ms min | — | 2.0 – 15.0 | | B0 field t | — | 1.5 – 7.0 | | Voxel size mm | mm | 0.5 – 2.0 | | B0 inhomogeneity | — | 0.0 – 0.3 | | Phase unwrap error | — | 0.0 – 0.3 | | Brain segmentation error | — | 0.0 – 0.2 | | Background field removal error | — | 0.0 – 0.3 | | Dipole kernel truncation error | — | 0.0 – 0.2 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 30.0 | Metric | Range | |---|---| | Psnr db | 10.0 – 50.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `240` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x5484b9c5e5758b3aa4e5899dc46d6d1ae08db76a89c897222e2c1d8d7f974782` - **Chain tx hash:** `0xbc7817f39f0ec56bc0fc58a6a8a6d43afe92dda05c0fd3d923f880b08273214a` - **Chain block:** `41553359` --- ## File Mapping This bundle consists of: `L1-503.md`, `L1-503.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-503.md` | Source of truth — edit this | Human or LLM | | `L1-503.json` | Structured metadata for the registry | LLM regenerates from the sections above | **Prompt for your LLM after editing this Markdown:** > Read the attached Markdown. Regenerate the sibling `.json` so every field matches. > Preserve the schema documented in the rows above. > Output each file in its own fenced code block tagged with the filename. > Output only the JSON object. _This Markdown was auto-synthesized from the catalog row for `L1-503`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-503.md).
Regenerate L1-503.json so every field matches the Markdown.
Preserve the schema documented in the File Mapping footer at the bottom of the MD.
Output each file in its own fenced code block tagged with the filename.

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L2-503-001 Quantitative Susceptibility Mapping (QSM) — Spec Skeleton

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📋 JSON metadata

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  "chain_tx_hash": "0xbc7817f39f0ec56bc0fc58a6a8a6d43afe92dda05c0fd3d923f880b08273214a",
  "domain": "Medical Imaging",
  "hardness_fn": {
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    "regime": "Existence of recovered 3D quantitative susceptibility chi(r) is guaranteed within the declared Omega bounds. Uniqueness holds on the wave-cone-supported subspace; modes inside the magic-angle zero cone are recovered only through priors (MEDI L1-edge, TGV second-order, deep-prior). Stability is moderately conditioned (kappa_eff ~ 30 after wave-cone regularization) \u2014 B0_inhomogeneity and background_field_removal_error dominate the stability cliff; phase_unwrap_error and dipole_kernel_truncation contribute higher-order bias terms. Joint Hadamard well-posedness for the coupled phase-field-susceptibility forward is established by Liu, de Rochefort, Wharton, Bilgic literature 2009-2018 (see references).",
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    "title": "Quantitative Susceptibility Mapping (QSM)"
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      "Z": [
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      ],
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    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Quantitative MR susceptibility recovery (multi-physics joint inverse)",
  "title": "Quantitative Susceptibility Mapping (QSM)"
}

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