# ⚛  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`

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## 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.
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_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._