# ⚛  L1 Principle — MR Spectroscopy Tumor Grading (PWDR)

**ID:** `L1-516` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Brain / prostate / breast tumor metabolite-ratio threshold classification from MR spectroscopy*
> **🎯 Problem class:** linear inverse with categorical readout · **🧮 Solution space:** 1D tumor grade
> **📡 Carrier:** radio_wave · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553373

---

## 🧠 1. Introduction

**MR Spectroscopy Tumor Grading (PWDR)** is a **linear inverse with categorical readout** whose unknown lives in **1D tumor grade** space, within the **Brain / prostate / breast tumor metabolite-ratio threshold classification from MR spectroscopy** sub-domain of **Medical Imaging**.

Measurements consist of radio-frequency electromagnetic waves via a **mrs chemical shift with grade threshold** sensing mechanism.

The forward operator applies, in order: Bloch-equation tip of the magnetisation vector; L · gradient echo operator; L · chemical shift select operator; L · metabolite quantification operator; L · ratio threshold classifier operator; detector sums all spectral bands; pixel-level spatial averaging on the detector.

Observations are corrupted by additive Gaussian noise. Existence and uniqueness inherited from L1-044 MRS. Stability inherits L1-044's kappa_eff plus additive contribution from voxel_lipid_contamination (dominant at TE<35 ms) and partial_volume_csf. Joint Hadamard well-posedness for the coupled MRS + metabolite-ratio threshold forward established by Howe 2003, Tate 2003, McKnight 2002, Kurhanewicz 2002 (prostate), Bolan 2003 (breast), Provencher 1993 (LCModel).

## ⚙ 2. Forward Model

Physical chain: **x** → RF excitation pulse → L · gradient echo → L · chemical shift select → L · metabolite quantification → L · ratio threshold classifier → Spectral integration → Spatial integration → **y** (detector).

```
y = ∫_A dA Σ_λ `L.ratio_threshold_classifier` `L.metabolite_quantification` `L.chemical_shift_select` `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.chemical_shift_select` | L · chemical shift select operator |
| `L.metabolite_quantification` | L · metabolite quantification operator |
| `L.ratio_threshold_classifier` | L · ratio threshold classifier operator |
| `int.spectral` | Detector sums all spectral bands |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Brain / prostate / breast tumor metabolite-ratio threshold classification from MR spectroscopy |
| Carrier | radio_wave |
| Problem class | linear_inverse_with_categorical_readout |
| Solution space | 1D_tumor_grade |
| Noise model | gaussian |
| Integration axis | spectral_spatial |
| Difficulty delta | 3 |
| L dag | 5.9 |

## 📡 4. Measurement Model

Existence and uniqueness inherited from L1-044 MRS. Stability inherits L1-044's kappa_eff plus additive contribution from voxel_lipid_contamination (dominant at TE<35 ms) and partial_volume_csf. Joint Hadamard well-posedness for the coupled MRS + metabolite-ratio threshold forward established by Howe 2003, Tate 2003, McKnight 2002, Kurhanewicz 2002 (prostate), Bolan 2003 (breast), Provencher 1993 (LCModel).

| Metric | Value |
|---|---|
| Metric | categorical_accuracy |
| Secondary | weighted_kappa_grade |

## 📏 5. Operating Range (Ω)

**Center problem class:** `mrs_brain_tumor_grading_pwdr` · **Forward operator:** `mrs_grade_pwdr_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Te ms | ms | 35 |
| Tr ms | ms | 2000 |
| Snr db | dB | 20 |
| H voxels | — | 16 |
| W voxels | — | 16 |
| Z voxels | — | 8 |
| B0 field t | — | 3 |
| Voxel size mm | mm | 8 |
| Spectral points | — | 1024 |
| B0 inhomogeneity | — | 0 |
| Partial volume csf | — | 0 |
| Voxel lipid contamination | — | 0 |
| Water suppression residual | — | 0 |
| Threshold calibration age effect | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Te ms | ms | 20 – 144 |
| H voxels | — | 4 – 64 |
| W voxels | — | 4 – 64 |
| Z voxels | — | 1 – 32 |
| B0 field t | — | 1.5 – 7.0 |
| Voxel size mm | mm | 3 – 20 |
| Spectral points | — | 256 – 4096 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.80_accuracy

| Metric | Range |
|---|---|
| Categorical accuracy | 0.5 – 0.95 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **categorical_accuracy**, with κ = `120` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0xd90de40fb6c4b8108d2c99e0497106abf5f24dfb4043a816f1b419791809f0e3`
- **Chain tx hash:** `0x3a2fe31113268dd36ab28f8e6dc824a30a6ab7606211529a395deecb669a1ebc`
- **Chain block:** `41553373`

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## File Mapping

This bundle consists of: `L1-516.md`, `L1-516.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-516.md` | Source of truth — edit this | Human or LLM |
| `L1-516.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 `L1-516`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._