L1 ⚛ L1-505 ⊙ Testnet

Pharmacokinetic Dynamic PET (PK-PET)

Medical Imaging · Quantitative tracer kinetic imaging (multi-physics joint inverse)

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⬇ L1-505.md

# ⚛ L1 Principle — Pharmacokinetic Dynamic PET (PK-PET) **ID:** `L1-505` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Quantitative tracer kinetic imaging (multi-physics joint inverse)* > **🎯 Problem class:** linear inverse 4d · **🧮 Solution space:** 4D kinetic parameter map > **📡 Carrier:** photon_511keV · **🌫 Noise:** poisson > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553386 --- ## 🧠 1. Introduction **Pharmacokinetic Dynamic PET (PK-PET)** is a **linear inverse 4d** whose unknown lives in **4D kinetic parameter map** space, within the **Quantitative tracer kinetic imaging (multi-physics joint inverse)** sub-domain of **Medical Imaging**. Measurements consist of photon 511keV via a **annihilation with compartmental kinetics** sensing mechanism. The forward operator applies, in order: L · tracer injection operator; L · compartmental ode operator; L · activity distribution operator; L · attenuation correction operator; L · scatter correction operator; L · lor projection operator; detector accumulates flux over the exposure window; pixel-level spatial averaging on the detector. Observations are corrupted by Poisson counting noise. Existence of recovered 4D kinetic parameter maps (K_1, k_2, k_3, k_4)(r) is guaranteed within the declared Omega bounds. Uniqueness holds for irreversible 2-tissue compartmental models (FDG-like, k_4 = 0) with sufficient temporal sampling and known C_p(t); reversible 2-tissue models (4 free parameters) require either reference-region constraint or sufficient SNR for full identifiability. Stability is moderately conditioned (kappa_eff ~ 25 after 4D-EM-ML or direct kinetic reconstruction) — input_function_uncertainty dominates K_1 bias; partial_volume_effect dominates small-structure quantitation; count_statistics_dropoff dominates late-frame variance. Joint Hadamard well-posedness for the coupled compartmental-PET forward is established by Carson (1996, 2003), Gunn-Gunn-Cunningham (2001), Patlak-Blasberg-Fenstermacher (1983 Patlak plot), Logan et al. (1990 Logan plot), Wang-Qi (2013 direct kinetic estimation), and Reader-Verhaeghe (2014 4D image reconstruction). ## ⚙ 2. Forward Model Physical chain: **x** → L · tracer injection → L · compartmental ode → L · activity distribution → L · attenuation correction → L · scatter correction → L · lor projection → Temporal integration → Spatial integration → **y** (detector). ``` y = ∫_A dA ∫_t dt `L.lor_projection` `L.scatter_correction` `L.attenuation_correction` `L.activity_distribution` `L.compartmental_ode` `L.tracer_injection` x, measurements ~ Poisson(αy) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.tracer_injection` | L · tracer injection operator | | `L.compartmental_ode` | L · compartmental ode operator | | `L.activity_distribution` | L · activity distribution operator | | `L.attenuation_correction` | L · attenuation correction operator | | `L.scatter_correction` | L · scatter correction operator | | `L.lor_projection` | L · lor projection operator | | `int.temporal` | Detector accumulates flux over the exposure window | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | Quantitative tracer kinetic imaging (multi-physics joint inverse) | | Carrier | photon_511keV | | Problem class | linear_inverse_4d | | Solution space | 4D_kinetic_parameter_map | | Noise model | poisson | | Integration axis | spatial_temporal | | Difficulty delta | 5 | | L dag | 8.3 | ## 📡 4. Measurement Model Existence of recovered 4D kinetic parameter maps (K_1, k_2, k_3, k_4)(r) is guaranteed within the declared Omega bounds. Uniqueness holds for irreversible 2-tissue compartmental models (FDG-like, k_4 = 0) with sufficient temporal sampling and known C_p(t); reversible 2-tissue models (4 free parameters) require either reference-region constraint or sufficient SNR for full identifiability. Stability is moderately conditioned (kappa_eff ~ 25 after 4D-EM-ML or direct kinetic reconstruction) — input_function_uncertainty dominates K_1 bias; partial_volume_effect dominates small-structure quantitation; count_statistics_dropoff dominates late-frame variance. Joint Hadamard well-posedness for the coupled compartmental-PET forward is established by Carson (1996, 2003), Gunn-Gunn-Cunningham (2001), Patlak-Blasberg-Fenstermacher (1983 Patlak plot), Logan et al. (1990 Logan plot), Wang-Qi (2013 direct kinetic estimation), and Reader-Verhaeghe (2014 4D image reconstruction). | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | RMSE_per_kinetic_parameter | ## 📏 5. Operating Range (Ω) **Center problem class:** `pkpet_2tissue_irreversible` · **Forward operator:** `pkpet_joint_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 256 | | W | px | 256 | | Z | — | 64 | | N frames | — | 28 | | Tracer class | — | FDG | | Voxel size mm | mm | 2 | | Compartment count | — | 2 | | Scan duration min | — | 60 | | Motion during scan | — | 0 | | Partial volume effect | — | 0 | | Count statistics dropoff | — | 0 | | Scatter correction error | — | 0 | | Input function uncertainty | — | 0 | | Noise equivalent count rate | — | 100000 | | Attenuation correction error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 128 – 512 | | W | px | 128 – 512 | | Z | — | 32 – 256 | | N frames | — | 12 – 60 | | Voxel size mm | mm | 1.0 – 4.0 | | Compartment count | — | 1 – 4 | | Scan duration min | — | 20 – 180 | | Motion during scan | — | 0.0 – 0.3 | | Partial volume effect | — | 0.0 – 0.4 | | Count statistics dropoff | — | 0.0 – 0.5 | | Scatter correction error | — | 0.0 – 0.2 | | Input function uncertainty | — | 0.0 – 0.3 | | Noise equivalent count rate | — | 10000 – 1000000 | | Attenuation correction error | — | 0.0 – 0.2 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 24.0 | Metric | Range | |---|---| | Psnr db | 8.0 – 42.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `200` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xe345d09b0e6f005469ab139b4deb1ae1ee8eccfac92ca8f8a188aec92f139809` - **Chain tx hash:** `0x2739c91ad7312bf16f884dc0b15f996df74fce8a13ed24bddd0871fcccca8779` - **Chain block:** `41553386` --- ## File Mapping This bundle consists of: `L1-505.md`, `L1-505.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-505.md` | Source of truth — edit this | Human or LLM | | `L1-505.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 `L1-505`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-505.md).
Regenerate L1-505.json so every field matches the Markdown.
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L2-505-001 Pharmacokinetic Dynamic PET (PK-PET) — Spec Skeleton

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📋 JSON metadata

{
  "artifact_id": "L1-505",
  "chain_block": 41553386,
  "chain_hash": "0xe345d09b0e6f005469ab139b4deb1ae1ee8eccfac92ca8f8a188aec92f139809",
  "chain_tx_hash": "0x2739c91ad7312bf16f884dc0b15f996df74fce8a13ed24bddd0871fcccca8779",
  "domain": "Medical Imaging",
  "hardness_fn": {
    "delta": 5,
    "kappa": 200,
    "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 4D kinetic parameter maps (K_1, k_2, k_3, k_4)(r) is guaranteed within the declared Omega bounds. Uniqueness holds for irreversible 2-tissue compartmental models (FDG-like, k_4 = 0) with sufficient temporal sampling and known C_p(t); reversible 2-tissue models (4 free parameters) require either reference-region constraint or sufficient SNR for full identifiability. Stability is moderately conditioned (kappa_eff ~ 25 after 4D-EM-ML or direct kinetic reconstruction) \u2014 input_function_uncertainty dominates K_1 bias; partial_volume_effect dominates small-structure quantitation; count_statistics_dropoff dominates late-frame variance. Joint Hadamard well-posedness for the coupled compartmental-PET forward is established by Carson (1996, 2003), Gunn-Gunn-Cunningham (2001), Patlak-Blasberg-Fenstermacher (1983 Patlak plot), Logan et al. (1990 Logan plot), Wang-Qi (2013 direct kinetic estimation), and Reader-Verhaeghe (2014 4D image reconstruction).",
    "secondary": "RMSE_per_kinetic_parameter"
  },
  "physics_fingerprint": {
    "L_DAG": 8.3,
    "carrier": "photon_511keV",
    "difficulty_delta": 5,
    "domain": "Medical Imaging",
    "integration_axis": "spatial_temporal",
    "noise_model": "poisson",
    "primitives": [
      "L.tracer_injection",
      "L.compartmental_ode",
      "L.activity_distribution",
      "L.attenuation_correction",
      "L.scatter_correction",
      "L.lor_projection",
      "int.temporal",
      "int.spatial"
    ],
    "problem_class": "linear_inverse_4d",
    "sensing_mechanism": "annihilation_with_compartmental_kinetics",
    "solution_space": "4D_kinetic_parameter_map",
    "sub_domain": "Quantitative tracer kinetic imaging (multi-physics joint inverse)",
    "title": "Pharmacokinetic Dynamic PET (PK-PET)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "pkpet_joint_forward",
      "pkpet_two_step_forward",
      "pkpet_direct_kinetic_forward",
      "pkpet_patlak_forward",
      "pkpet_logan_forward"
    ],
    "allowed_omega_dimensions": [
      "H",
      "W",
      "Z",
      "N_frames",
      "scan_duration_min",
      "voxel_size_mm",
      "tracer_class",
      "compartment_count",
      "noise_equivalent_count_rate",
      "input_function_uncertainty",
      "attenuation_correction_error",
      "scatter_correction_error",
      "motion_during_scan",
      "partial_volume_effect",
      "count_statistics_dropoff"
    ],
    "allowed_problem_classes": [
      "pkpet_2tissue_irreversible",
      "pkpet_2tissue_reversible",
      "pkpet_3tissue_metabolite",
      "pkpet_reference_tissue",
      "pkpet_graphical_irreversible",
      "pkpet_graphical_reversible"
    ],
    "center_spec": {
      "epsilon_fn_center": "24.0",
      "forward_operator": "pkpet_joint_forward",
      "input_format": "dynamic_sinogram_with_input_function",
      "omega": {
        "H": 256,
        "N_frames": 28,
        "W": 256,
        "Z": 64,
        "attenuation_correction_error": 0.0,
        "compartment_count": 2,
        "count_statistics_dropoff": 0.0,
        "input_function_uncertainty": 0.0,
        "motion_during_scan": 0.0,
        "noise_equivalent_count_rate": 100000,
        "partial_volume_effect": 0.0,
        "scan_duration_min": 60,
        "scatter_correction_error": 0.0,
        "tracer_class": "FDG",
        "voxel_size_mm": 2.0
      },
      "problem_class": "pkpet_2tissue_irreversible"
    },
    "epsilon_bounds": {
      "psnr_db": [
        8.0,
        42.0
      ]
    },
    "omega_bounds": {
      "H": [
        128,
        512
      ],
      "N_frames": [
        12,
        60
      ],
      "W": [
        128,
        512
      ],
      "Z": [
        32,
        256
      ],
      "attenuation_correction_error": [
        0.0,
        0.2
      ],
      "compartment_count": [
        1,
        4
      ],
      "count_statistics_dropoff": [
        0.0,
        0.5
      ],
      "input_function_uncertainty": [
        0.0,
        0.3
      ],
      "motion_during_scan": [
        0.0,
        0.3
      ],
      "noise_equivalent_count_rate": [
        10000,
        1000000
      ],
      "partial_volume_effect": [
        0.0,
        0.4
      ],
      "scan_duration_min": [
        20,
        180
      ],
      "scatter_correction_error": [
        0.0,
        0.2
      ],
      "voxel_size_mm": [
        1.0,
        4.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Quantitative tracer kinetic imaging (multi-physics joint inverse)",
  "title": "Pharmacokinetic Dynamic PET (PK-PET)"
}

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