L1 ⚛ L1-506 ⊙ Testnet

Hyperpolarized 13C Metabolic MRI

Medical Imaging · Real-time metabolic kinetic imaging via dynamic nuclear polarization (multi-physics joint inverse)

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⬇ L1-506.md

# ⚛ L1 Principle — Hyperpolarized 13C Metabolic MRI **ID:** `L1-506` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Real-time metabolic kinetic imaging via dynamic nuclear polarization (multi-physics joint inverse)* > **🎯 Problem class:** nonlinear inverse 4d · **🧮 Solution space:** 4D metabolic rate constant map > **📡 Carrier:** spin_13C_hyperpolarized · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41552304 --- ## 🧠 1. Introduction **Hyperpolarized 13C Metabolic MRI** is a **nonlinear inverse 4d** whose unknown lives in **4D metabolic rate constant map** space, within the **Real-time metabolic kinetic imaging via dynamic nuclear polarization (multi-physics joint inverse)** sub-domain of **Medical Imaging**. Measurements consist of spin 13C hyperpolarized via a **dnp with chemical exchange and chemical shift mri** sensing mechanism. The forward operator applies, in order: L · dnp polarization operator; L · injection bolus operator; L · chemical exchange ode operator; L · bloch dynamics operator; Bloch-equation tip of the magnetisation vector; L · spectral spatial encoding operator; L · gradient readout operator; detector sums all spectral bands; detector accumulates flux over the exposure window. Observations are corrupted by additive Gaussian noise. Existence of recovered metabolic rate-constant maps (k_PL, k_PA, k_PB)(r) and per-pool T1 maps is guaranteed within the declared Omega bounds. Uniqueness holds when at least 3 metabolite pools are sampled with sufficient temporal resolution (frame duration <= 5 s) and SNR per metabolite > 10 dB; degenerate cases (all rate constants near zero, or T1 << acquisition window) require regularization. Stability is moderately conditioned (kappa_eff ~ 50 after rate-constant-aware spatiotemporal regularization) — dnp_polarization_loss dominates absolute rate-constant scaling; injection_bolus_uncertainty dominates K_1-equivalent bias; t1_uncertainty contributes a per-pool scaling factor; partial_volume_effect dominates small-structure quantitation. Joint Hadamard well-posedness for the coupled DNP-chemical-exchange-MR forward is established by Bahrami et al. (2014), Larson et al. (2018), Maidens-Gordon-Arcak (2016 control-theoretic identifiability), and Brindle (2015 review on hyperpolarized MR imaging principles). ## ⚙ 2. Forward Model Physical chain: **x** → L · dnp polarization → L · injection bolus → L · chemical exchange ode → L · bloch dynamics → RF excitation pulse → L · spectral spatial encoding → L · gradient readout → Spectral integration → Temporal integration → **y** (detector). ``` y = ∫_t dt Σ_λ `L.gradient_readout` `L.spectral_spatial_encoding` B₁(t) `L.bloch_dynamics` `L.chemical_exchange_ode` `L.injection_bolus` `L.dnp_polarization` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.dnp_polarization` | L · dnp polarization operator | | `L.injection_bolus` | L · injection bolus operator | | `L.chemical_exchange_ode` | L · chemical exchange ode operator | | `L.bloch_dynamics` | L · bloch dynamics operator | | `L.rf_excitation` | Bloch-equation tip of the magnetisation vector | | `L.spectral_spatial_encoding` | L · spectral spatial encoding operator | | `L.gradient_readout` | L · gradient readout operator | | `int.spectral` | Detector sums all spectral bands | | `int.temporal` | Detector accumulates flux over the exposure window | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | Real-time metabolic kinetic imaging via dynamic nuclear polarization (multi-physics joint inverse) | | Carrier | spin_13C_hyperpolarized | | Problem class | nonlinear_inverse_4d | | Solution space | 4D_metabolic_rate_constant_map | | Noise model | gaussian | | Integration axis | spectral_temporal_spatial | | Difficulty delta | 5 | | L dag | 10.4 | ## 📡 4. Measurement Model Existence of recovered metabolic rate-constant maps (k_PL, k_PA, k_PB)(r) and per-pool T1 maps is guaranteed within the declared Omega bounds. Uniqueness holds when at least 3 metabolite pools are sampled with sufficient temporal resolution (frame duration <= 5 s) and SNR per metabolite > 10 dB; degenerate cases (all rate constants near zero, or T1 << acquisition window) require regularization. Stability is moderately conditioned (kappa_eff ~ 50 after rate-constant-aware spatiotemporal regularization) — dnp_polarization_loss dominates absolute rate-constant scaling; injection_bolus_uncertainty dominates K_1-equivalent bias; t1_uncertainty contributes a per-pool scaling factor; partial_volume_effect dominates small-structure quantitation. Joint Hadamard well-posedness for the coupled DNP-chemical-exchange-MR forward is established by Bahrami et al. (2014), Larson et al. (2018), Maidens-Gordon-Arcak (2016 control-theoretic identifiability), and Brindle (2015 review on hyperpolarized MR imaging principles). | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | RMSE_per_rate_constant | ## 📏 5. Operating Range (Ω) **Center problem class:** `hp13c_pyruvate_3pool` · **Forward operator:** `hp13c_joint_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 64 | | W | px | 64 | | Z | — | 16 | | P dnp | — | 0.3 | | Snr db | dB | 18 | | N frames | — | 20 | | B0 field t | — | 3 | | N metabolites | — | 3 | | Voxel size mm | mm | 5 | | T1 uncertainty | — | 0 | | B1 inhomogeneity | — | 0 | | Frame duration s | s | 3 | | Dnp polarization loss | — | 0 | | Partial volume effect | — | 0 | | Susceptibility artifact | — | 0 | | Injection bolus uncertainty | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 32 – 256 | | W | px | 32 – 256 | | Z | — | 8 – 64 | | P dnp | — | 0.05 – 0.5 | | Snr db | dB | 5.0 – 35.0 | | N frames | — | 5 – 60 | | B0 field t | — | 1.5 – 7.0 | | N metabolites | — | 2 – 6 | | Voxel size mm | mm | 2.0 – 15.0 | | T1 uncertainty | — | 0.0 – 0.3 | | B1 inhomogeneity | — | 0.0 – 0.3 | | Frame duration s | s | 1.0 – 10.0 | | Dnp polarization loss | — | 0.0 – 0.4 | | Partial volume effect | — | 0.0 – 0.4 | | Susceptibility artifact | — | 0.0 – 0.3 | | Injection bolus uncertainty | — | 0.0 – 0.3 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 22.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 40.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `280` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x97ad824ebd08b02b8f9189ec95f881e277530e6417b27853ca4157303a266ce8` - **Chain tx hash:** `0xdb7bb7e798e0ff9cffb9dd27b33714c756c74b69b9a179180d86a48f7a70a4a6` - **Chain block:** `41552304` --- ## File Mapping This bundle consists of: `L1-506.md`, `L1-506.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-506.md` | Source of truth — edit this | Human or LLM | | `L1-506.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 `L1-506`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Regenerate L1-506.json so every field matches the Markdown.
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L2-506-001 Hyperpolarized 13C Metabolic MRI — Spec Skeleton

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📋 JSON metadata

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    "regime": "Existence of recovered metabolic rate-constant maps (k_PL, k_PA, k_PB)(r) and per-pool T1 maps is guaranteed within the declared Omega bounds. Uniqueness holds when at least 3 metabolite pools are sampled with sufficient temporal resolution (frame duration \u003c= 5 s) and SNR per metabolite \u003e 10 dB; degenerate cases (all rate constants near zero, or T1 \u003c\u003c acquisition window) require regularization. Stability is moderately conditioned (kappa_eff ~ 50 after rate-constant-aware spatiotemporal regularization) \u2014 dnp_polarization_loss dominates absolute rate-constant scaling; injection_bolus_uncertainty dominates K_1-equivalent bias; t1_uncertainty contributes a per-pool scaling factor; partial_volume_effect dominates small-structure quantitation. Joint Hadamard well-posedness for the coupled DNP-chemical-exchange-MR forward is established by Bahrami et al. (2014), Larson et al. (2018), Maidens-Gordon-Arcak (2016 control-theoretic identifiability), and Brindle (2015 review on hyperpolarized MR imaging principles).",
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  "sub_domain": "Real-time metabolic kinetic imaging via dynamic nuclear polarization (multi-physics joint inverse)",
  "title": "Hyperpolarized 13C Metabolic MRI"
}

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