# ⚛  L1 Principle — Mammographic Breast Density Classification BI-RADS (PWDR)

**ID:** `L1-515` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Mammographic glandular-tissue density reconstruction with BI-RADS density category readout*
> **🎯 Problem class:** linear inverse with categorical readout · **🧮 Solution space:** 1D birads density category
> **📡 Carrier:** x_ray · **🌫 Noise:** poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553360

---

## 🧠 1. Introduction

**Mammographic Breast Density Classification BI-RADS (PWDR)** is a **linear inverse with categorical readout** whose unknown lives in **1D birads density category** space, within the **Mammographic glandular-tissue density reconstruction with BI-RADS density category readout** sub-domain of **Medical Imaging**.

Measurements consist of X-ray photons transmitted through (or scattered by) the sample via a **mammography with birads density grading** sensing mechanism.

The forward operator applies, in order: polyenergetic X-ray emission spectrum; line-integral projection through an attenuation map; L · dual energy decomposition operator; L · glandular fraction recovery operator; L · vbd aggregation operator; L · birads threshold classifier operator; pixel-level spatial averaging on the detector.

Observations are corrupted by Poisson counting noise. Existence and uniqueness inherited from L1-036 mammography. Stability inherits L1-036's well-conditioned reconstruction plus a small additive contribution from compression_thickness_calibration_error (dominant systematic bias source) and age_related_density_shift. Joint Hadamard well-posedness for the coupled mammography + BI-RADS threshold forward established by Sickles E et al. (2013 BI-RADS Atlas 5th ed), Highnam R et al. (2010 Volpara volumetric density), Hartman K et al. (2013 Quantra), Yaffe MJ (2008 review of mammographic density), Boyd NF et al. (2007 density and breast cancer risk), and Pertuz S et al. (2014 automated density estimation comparison).

## ⚙ 2. Forward Model

Physical chain: **x** → X-ray source → Attenuation projection → L · dual energy decomposition → L · glandular fraction recovery → L · vbd aggregation → L · birads threshold classifier → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.birads_threshold_classifier` `L.vbd_aggregation` `L.glandular_fraction_recovery` `L.dual_energy_decomposition` ∫µ dl I₀(E) x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.xray_source` | Polyenergetic x-ray emission spectrum |
| `L.attenuation_projection` | Line-integral projection through an attenuation map |
| `L.dual_energy_decomposition` | L · dual energy decomposition operator |
| `L.glandular_fraction_recovery` | L · glandular fraction recovery operator |
| `L.vbd_aggregation` | L · vbd aggregation operator |
| `L.birads_threshold_classifier` | L · birads threshold classifier operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Mammographic glandular-tissue density reconstruction with BI-RADS density category readout |
| Carrier | x_ray |
| Problem class | linear_inverse_with_categorical_readout |
| Solution space | 1D_birads_density_category |
| Noise model | poisson |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 5.8 |

## 📡 4. Measurement Model

Existence and uniqueness inherited from L1-036 mammography. Stability inherits L1-036's well-conditioned reconstruction plus a small additive contribution from compression_thickness_calibration_error (dominant systematic bias source) and age_related_density_shift. Joint Hadamard well-posedness for the coupled mammography + BI-RADS threshold forward established by Sickles E et al. (2013 BI-RADS Atlas 5th ed), Highnam R et al. (2010 Volpara volumetric density), Hartman K et al. (2013 Quantra), Yaffe MJ (2008 review of mammographic density), Boyd NF et al. (2007 density and breast cancer risk), and Pertuz S et al. (2014 automated density estimation comparison).

| Metric | Value |
|---|---|
| Metric | categorical_accuracy |
| Secondary | weighted_kappa_birads |

## 📏 5. Operating Range (Ω)

**Center problem class:** `mammographic_birads_density_pwdr` · **Forward operator:** `mammography_birads_pwdr_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Kvp | — | 30 |
| Mas | — | 100 |
| H proj | — | 3500 |
| Snr db | dB | 32 |
| W proj | — | 2800 |
| Pixel size um | µm | 85 |
| Filter material | — | Mo |
| Anode target material | — | Mo |
| Exposure factor drift | — | 0 |
| Compression thickness mm | mm | 50 |
| Age related density shift | — | 0 |
| Calibration phantom uncertainty | — | 0 |
| Automated vs radiologist disagreement | — | 0 |
| Compression thickness calibration error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Kvp | — | 25 – 35 |
| H proj | — | 1024 – 5500 |
| W proj | — | 1024 – 4500 |
| Pixel size um | µm | 50 – 200 |
| Compression thickness mm | mm | 20 – 90 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.85_accuracy

| Metric | Range |
|---|---|
| Categorical accuracy | 0.5 – 0.99 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **categorical_accuracy**, with κ = `60` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xbc0db9931caa3bf1b6cd65d243103412660149215ca9cf038c3720ecf6ae0c52`
- **Chain tx hash:** `0xa728d0236a9dc4d96f2b2a3ff9d3dd2088fa9583d6ebecfb8ca1a2803e52f838`
- **Chain block:** `41553360`

---

## File Mapping

This bundle consists of: `L1-515.md`, `L1-515.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-515.md` | Source of truth — edit this | Human or LLM |
| `L1-515.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 `L1-515`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._