# ⚛  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

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## 🧠 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`

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## 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.
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> 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._