L1 ⚛ L1-504 ⊙ Testnet

Quantitative Photoacoustic Tomography (qPAT)

Medical Imaging · Quantitative multi-spectral chromophore recovery (multi-physics joint inverse)

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⬇ L1-504.md

# ⚛ L1 Principle — Quantitative Photoacoustic Tomography (qPAT) **ID:** `L1-504` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Quantitative multi-spectral chromophore recovery (multi-physics joint inverse)* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D spectral chromophore quantitative > **📡 Carrier:** photon_to_acoustic · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553373 --- ## 🧠 1. Introduction **Quantitative Photoacoustic Tomography (qPAT)** is a **nonlinear inverse problem** whose unknown lives in **3D spectral chromophore quantitative** space, within the **Quantitative multi-spectral chromophore recovery (multi-physics joint inverse)** sub-domain of **Medical Imaging**. Measurements consist of photon to acoustic via a **thermoelastic with optical diffusion** sensing mechanism. The forward operator applies, in order: L · optical source operator; L · optical diffusion operator; exponential attenuation along the propagation path; L · thermoelastic grueneisen operator; L · acoustic wave propagation operator; L · transducer detection operator; detector accumulates flux over the exposure window; detector sums all spectral bands. Observations are corrupted by additive Gaussian noise. Existence of recovered 3D multi-spectral absorption coefficient mu_a(r, lambda) is guaranteed within the declared Omega bounds. Uniqueness holds under multi-illumination or multi-wavelength acquisition (Bal-Uhlmann 2010; Bal-Ren 2011); single-illumination single-wavelength qPAT is non-unique due to the multiplicative H = mu_a * Phi coupling and is excluded from the spec range. Stability is moderately conditioned (kappa_eff ~ 40 after model-based reconstruction) — acoustic bandwidth dominates spatial resolution; mu_s_prime_uncertainty dominates absorption-coefficient bias; gruneisen_uncertainty contributes a scaling factor. Joint Hadamard well-posedness for the coupled optical-thermoelastic-acoustic forward is established by Cox-Arridge-Beard (2009), Bal-Uhlmann (2010), Bal-Ren (2011), Tarvainen et al. (2013), and Cox-Tarvainen-Arridge (2014). See joint_well_posedness_references. ## ⚙ 2. Forward Model Physical chain: **x** → L · optical source → L · optical diffusion → Beer-Lambert attenuation → L · thermoelastic grueneisen → L · acoustic wave propagation → L · transducer detection → Temporal integration → Spectral integration → **y** (detector). ``` y = Σ_λ ∫_t dt `L.transducer_detection` `L.acoustic_wave_propagation` `L.thermoelastic_grueneisen` exp(-∫µ dl) `L.optical_diffusion` `L.optical_source` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.optical_source` | L · optical source operator | | `L.optical_diffusion` | L · optical diffusion operator | | `L.beer_lambert` | Exponential attenuation along the propagation path | | `L.thermoelastic_grueneisen` | L · thermoelastic grueneisen operator | | `L.acoustic_wave_propagation` | L · acoustic wave propagation operator | | `L.transducer_detection` | L · transducer detection operator | | `int.temporal` | Detector accumulates flux over the exposure window | | `int.spectral` | Detector sums all spectral bands | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | Quantitative multi-spectral chromophore recovery (multi-physics joint inverse) | | Carrier | photon_to_acoustic | | Problem class | nonlinear_inverse | | Solution space | 3D_spectral_chromophore_quantitative | | Noise model | gaussian | | Integration axis | spectral_temporal_spatial | | Difficulty delta | 5 | | L dag | 8.5 | ## 📡 4. Measurement Model Existence of recovered 3D multi-spectral absorption coefficient mu_a(r, lambda) is guaranteed within the declared Omega bounds. Uniqueness holds under multi-illumination or multi-wavelength acquisition (Bal-Uhlmann 2010; Bal-Ren 2011); single-illumination single-wavelength qPAT is non-unique due to the multiplicative H = mu_a * Phi coupling and is excluded from the spec range. Stability is moderately conditioned (kappa_eff ~ 40 after model-based reconstruction) — acoustic bandwidth dominates spatial resolution; mu_s_prime_uncertainty dominates absorption-coefficient bias; gruneisen_uncertainty contributes a scaling factor. Joint Hadamard well-posedness for the coupled optical-thermoelastic-acoustic forward is established by Cox-Arridge-Beard (2009), Bal-Uhlmann (2010), Bal-Ren (2011), Tarvainen et al. (2013), and Cox-Tarvainen-Arridge (2014). See joint_well_posedness_references. | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | RMSE_per_chromophore | ## 📏 5. Operating Range (Ω) **Center problem class:** `qpat` · **Forward operator:** `qpat_joint_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 256 | | W | px | 256 | | Z | — | 64 | | Snr db | dB | 22 | | N wavelengths | — | 5 | | Voxel size mm | mm | 0.2 | | Lambda range nm | nm | 700 – 900 | | Speed of sound mps | — | 1500 | | Gruneisen uncertainty | — | 0 | | Mu s prime uncertainty | — | 0 | | Transducer bandwidth mhz | MHz | 5 | | Transducer position error | — | 0 | | Background absorption drift | — | 0 | | Light fluence inhomogeneity | — | 0 | | Acoustic speed heterogeneity | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 128 – 512 | | W | px | 128 – 512 | | Z | — | 32 – 256 | | Snr db | dB | 5.0 – 40.0 | | N wavelengths | — | 2 – 16 | | Voxel size mm | mm | 0.05 – 1.0 | | Lambda range nm | nm | 600 – 1100 | | Speed of sound mps | — | 1400.0 – 1600.0 | | Gruneisen uncertainty | — | 0.0 – 0.3 | | Mu s prime uncertainty | — | 0.0 – 0.3 | | Transducer bandwidth mhz | MHz | 1.0 – 30.0 | | Transducer position error | — | 0.0 – 0.2 | | Background absorption drift | — | 0.0 – 0.2 | | Light fluence inhomogeneity | — | 0.0 – 0.4 | | Acoustic speed heterogeneity | — | 0.0 – 0.1 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 26.0 | Metric | Range | |---|---| | Psnr db | 8.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `300` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xc92428e1d5e53fde3daba7c22ca0b6878713dd81dc0efe3ece43825b2119c710` - **Chain tx hash:** `0x766d26c2ec52e79e19fa7f1da0fa4773c85019d2cc461669a8dacd7962f2adfb` - **Chain block:** `41553373` --- ## File Mapping This bundle consists of: `L1-504.md`, `L1-504.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-504.md` | Source of truth — edit this | Human or LLM | | `L1-504.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 `L1-504`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-504.md).
Regenerate L1-504.json so every field matches the Markdown.
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L2-504-001 Quantitative Photoacoustic Tomography (qPAT) — Spec Skeleton

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📋 JSON metadata

{
  "artifact_id": "L1-504",
  "chain_block": 41553373,
  "chain_hash": "0xc92428e1d5e53fde3daba7c22ca0b6878713dd81dc0efe3ece43825b2119c710",
  "chain_tx_hash": "0x766d26c2ec52e79e19fa7f1da0fa4773c85019d2cc461669a8dacd7962f2adfb",
  "domain": "Medical Imaging",
  "hardness_fn": {
    "delta": 5,
    "kappa": 300,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "PSNR_dB",
    "regime": "Existence of recovered 3D multi-spectral absorption coefficient mu_a(r, lambda) is guaranteed within the declared Omega bounds. Uniqueness holds under multi-illumination or multi-wavelength acquisition (Bal-Uhlmann 2010; Bal-Ren 2011); single-illumination single-wavelength qPAT is non-unique due to the multiplicative H = mu_a * Phi coupling and is excluded from the spec range. Stability is moderately conditioned (kappa_eff ~ 40 after model-based reconstruction) \u2014 acoustic bandwidth dominates spatial resolution; mu_s_prime_uncertainty dominates absorption-coefficient bias; gruneisen_uncertainty contributes a scaling factor. Joint Hadamard well-posedness for the coupled optical-thermoelastic-acoustic forward is established by Cox-Arridge-Beard (2009), Bal-Uhlmann (2010), Bal-Ren (2011), Tarvainen et al. (2013), and Cox-Tarvainen-Arridge (2014). See joint_well_posedness_references.",
    "secondary": "RMSE_per_chromophore"
  },
  "physics_fingerprint": {
    "L_DAG": 8.5,
    "carrier": "photon_to_acoustic",
    "difficulty_delta": 5,
    "domain": "Medical Imaging",
    "integration_axis": "spectral_temporal_spatial",
    "noise_model": "gaussian",
    "primitives": [
      "L.optical_source",
      "L.optical_diffusion",
      "L.beer_lambert",
      "L.thermoelastic_grueneisen",
      "L.acoustic_wave_propagation",
      "L.transducer_detection",
      "int.temporal",
      "int.spectral"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "thermoelastic_with_optical_diffusion",
    "solution_space": "3D_spectral_chromophore_quantitative",
    "sub_domain": "Quantitative multi-spectral chromophore recovery (multi-physics joint inverse)",
    "title": "Quantitative Photoacoustic Tomography (qPAT)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "qpat_joint_forward",
      "qpat_two_step_forward",
      "qpat_multi_illumination_forward",
      "qpat_diffusion_approx_forward",
      "qpat_radiative_transfer_forward"
    ],
    "allowed_omega_dimensions": [
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      "W",
      "Z",
      "N_wavelengths",
      "lambda_range_nm",
      "voxel_size_mm",
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      "transducer_bandwidth_MHz",
      "SNR_dB",
      "mu_s_prime_uncertainty",
      "gruneisen_uncertainty",
      "acoustic_speed_heterogeneity",
      "transducer_position_error",
      "light_fluence_inhomogeneity",
      "background_absorption_drift"
    ],
    "allowed_problem_classes": [
      "qpat",
      "qpat_multi_wavelength",
      "qpat_multi_illumination",
      "qpat_chromophore_unmixing"
    ],
    "center_spec": {
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      "forward_operator": "qpat_joint_forward",
      "input_format": "multi_wavelength_time_resolved_pressure",
      "omega": {
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        "N_wavelengths": 5,
        "SNR_dB": 22,
        "W": 256,
        "Z": 64,
        "acoustic_speed_heterogeneity": 0.0,
        "background_absorption_drift": 0.0,
        "gruneisen_uncertainty": 0.0,
        "lambda_range_nm": [
          700,
          900
        ],
        "light_fluence_inhomogeneity": 0.0,
        "mu_s_prime_uncertainty": 0.0,
        "speed_of_sound_mps": 1500.0,
        "transducer_bandwidth_MHz": 5.0,
        "transducer_position_error": 0.0,
        "voxel_size_mm": 0.2
      },
      "problem_class": "qpat"
    },
    "epsilon_bounds": {
      "psnr_db": [
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        45.0
      ]
    },
    "omega_bounds": {
      "H": [
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        512
      ],
      "N_wavelengths": [
        2,
        16
      ],
      "SNR_dB": [
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        40.0
      ],
      "W": [
        128,
        512
      ],
      "Z": [
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        256
      ],
      "acoustic_speed_heterogeneity": [
        0.0,
        0.1
      ],
      "background_absorption_drift": [
        0.0,
        0.2
      ],
      "gruneisen_uncertainty": [
        0.0,
        0.3
      ],
      "lambda_range_nm": [
        600,
        1100
      ],
      "light_fluence_inhomogeneity": [
        0.0,
        0.4
      ],
      "mu_s_prime_uncertainty": [
        0.0,
        0.3
      ],
      "speed_of_sound_mps": [
        1400.0,
        1600.0
      ],
      "transducer_bandwidth_MHz": [
        1.0,
        30.0
      ],
      "transducer_position_error": [
        0.0,
        0.2
      ],
      "voxel_size_mm": [
        0.05,
        1.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Quantitative multi-spectral chromophore recovery (multi-physics joint inverse)",
  "title": "Quantitative Photoacoustic Tomography (qPAT)"
}

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