L1 ⚛ L1-515 ⊙ Testnet

Mammographic Breast Density Classification BI-RADS (PWDR)

Medical Imaging · Mammographic glandular-tissue density reconstruction with BI-RADS density category readout

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⬇ L1-515.md

# ⚛ 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._

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Read the attached Markdown (L1-515.md).
Regenerate L1-515.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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L2-515-001 Mammographic Breast Density Classification BI-RADS (PWDR) — Spec Skeleton

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📋 JSON metadata

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  "chain_block": 41553360,
  "chain_hash": "0xbc0db9931caa3bf1b6cd65d243103412660149215ca9cf038c3720ecf6ae0c52",
  "chain_tx_hash": "0xa728d0236a9dc4d96f2b2a3ff9d3dd2088fa9583d6ebecfb8ca1a2803e52f838",
  "domain": "Medical Imaging",
  "hardness_fn": {
    "delta": 3,
    "kappa": 60,
    "metric": "categorical_accuracy",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
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      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "categorical_accuracy",
    "regime": "Existence and uniqueness inherited from L1-036 mammography. Stability inherits L1-036\u0027s 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).",
    "secondary": "weighted_kappa_birads"
  },
  "physics_fingerprint": {
    "L_DAG": 5.8,
    "carrier": "x_ray",
    "difficulty_delta": 3,
    "domain": "Medical Imaging",
    "integration_axis": "spatial",
    "noise_model": "poisson",
    "primitives": [
      "L.xray_source",
      "L.attenuation_projection",
      "L.dual_energy_decomposition",
      "L.glandular_fraction_recovery",
      "L.vbd_aggregation",
      "L.birads_threshold_classifier",
      "int.spatial"
    ],
    "problem_class": "linear_inverse_with_categorical_readout",
    "sensing_mechanism": "mammography_with_birads_density_grading",
    "solution_space": "1D_birads_density_category",
    "sub_domain": "Mammographic glandular-tissue density reconstruction with BI-RADS density category readout",
    "title": "Mammographic Breast Density Classification BI-RADS (PWDR)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "mammography_birads_pwdr_forward",
      "dbt_birads_pwdr_forward",
      "ffdm_volpara_forward",
      "ffdm_quantra_forward"
    ],
    "allowed_omega_dimensions": [
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      "pixel_size_um",
      "kVp",
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      "anode_target_material",
      "filter_material",
      "SNR_dB",
      "compression_thickness_calibration_error",
      "exposure_factor_drift",
      "calibration_phantom_uncertainty",
      "age_related_density_shift",
      "automated_vs_radiologist_disagreement"
    ],
    "allowed_problem_classes": [
      "mammographic_birads_density_pwdr",
      "mammographic_4cat_density_pwdr",
      "mammographic_continuous_vbd"
    ],
    "center_spec": {
      "epsilon_fn_center": "0.85_accuracy",
      "forward_operator": "mammography_birads_pwdr_forward",
      "input_format": "ffdm_4view_with_density_label",
      "omega": {
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        "SNR_dB": 32,
        "W_proj": 2800,
        "age_related_density_shift": 0.0,
        "anode_target_material": "Mo",
        "automated_vs_radiologist_disagreement": 0.0,
        "calibration_phantom_uncertainty": 0.0,
        "compression_thickness_calibration_error": 0.0,
        "compression_thickness_mm": 50,
        "exposure_factor_drift": 0.0,
        "filter_material": "Mo",
        "kVp": 30,
        "mAs": 100,
        "pixel_size_um": 85
      },
      "problem_class": "mammographic_birads_density_pwdr"
    },
    "epsilon_bounds": {
      "categorical_accuracy": [
        0.5,
        0.99
      ]
    },
    "omega_bounds": {
      "H_proj": [
        1024,
        5500
      ],
      "W_proj": [
        1024,
        4500
      ],
      "compression_thickness_mm": [
        20,
        90
      ],
      "kVp": [
        25,
        35
      ],
      "pixel_size_um": [
        50,
        200
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Mammographic glandular-tissue density reconstruction with BI-RADS density category readout",
  "title": "Mammographic Breast Density Classification BI-RADS (PWDR)"
}

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