# ⚛  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._