# ⚛  L1 Principle — Physiologically Based PK (PBPK)

**ID:** `L1-403` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Systems pharmacology*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** PBPK parameter vector
> **📡 Carrier:** N/A · **🌫 Noise:** lognormal
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555218

---

## 🧠 1. Introduction

**Physiologically Based PK (PBPK)** is a **parameter-estimation problem** whose unknown lives in **PBPK parameter vector** space, within the **Systems pharmacology** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **tissue concentration measurement** sensing mechanism.

The forward operator applies, in order: time evolution of the state; S · qsr · tissue binding operator; O · nlme · pbpk fit operator.

Observations are corrupted by log-normal multiplicative noise. Existence of the recovered PBPK_parameter_vector is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 500); in_vitro_in_vivo_extrapolation_IVIVE_bias dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Lognormal sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Time derivative → S · qsr · tissue binding → O · nlme · pbpk fit → **y** (detector).

```
y = `O.nlme.pbpk_fit` `S.qsr.tissue_binding` ∂_t x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.time` | Time evolution of the state |
| `S.qsr.tissue_binding` | S · qsr · tissue binding operator |
| `O.nlme.pbpk_fit` | O · nlme · pbpk fit operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Systems pharmacology |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | PBPK_parameter_vector |
| Noise model | lognormal |
| Integration axis | organ_system |
| Difficulty delta | 5 |
| L dag | 4 |

## 📡 4. Measurement Model

Existence of the recovered PBPK_parameter_vector is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 500); in_vitro_in_vivo_extrapolation_IVIVE_bias dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Lognormal sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | plasma_RMSE_fold_error |
| Secondary | tissue_Kp_RMSE_log |

## 📏 5. Operating Range (Ω)

**Center problem class:** `parameter_estimation` · **Forward operator:** `tissue_concentration_measurement`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Cl int l hr | — | 20 |
| Species bw kg | kg | 70 |
| N organs modeled | — | 15 |
| Fu fraction unbound | — | 0.1 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Cl int l hr | — | 0.1 – 1000 |
| Species bw kg | kg | 0.02 – 500 |
| N organs modeled | — | 5 – 25 |
| Fu fraction unbound | — | 0.001 – 1.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 2 plasma_RMSE_fold_error

| Metric | Range |
|---|---|
| Plasma rmse fold error | 1.0 – 10.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **plasma_RMSE_fold_error**, with κ = `10000.0` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x01a1cd45de8d4c7fffff836b170d1e55dc5e0d7886115b617875f91fae49aa8b`
- **Chain tx hash:** `0x3413d775b5969bb353b1f6a4ebc170167950d11d98a7eac9217ed5de6acf3831`
- **Chain block:** `41555218`

---

## File Mapping

This bundle consists of: `L1-403.md`, `L1-403.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-403.md` | Source of truth — edit this | Human or LLM |
| `L1-403.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 `L1-403`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._