# ⚛  L1 Principle — Tumor Growth Model Inversion

**ID:** `L1-410` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Cancer modeling*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** tumor growth parameter vector
> **📡 Carrier:** N/A · **🌫 Noise:** lognormal
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555219

---

## 🧠 1. Introduction

**Tumor Growth Model Inversion** is a **parameter-estimation problem** whose unknown lives in **tumor growth parameter vector** space, within the **Cancer modeling** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **tumor volume imaging** sensing mechanism.

The forward operator applies, in order: time evolution of the state; O · nls · tumor fit operator; S · bootstrap · uncertainty tumor operator.

Observations are corrupted by log-normal multiplicative noise. Existence of the recovered tumor_growth_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 ~= 20); imaging_noise_percent 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 → O · nls · tumor fit → S · bootstrap · uncertainty tumor → **y** (detector).

```
y = `S.bootstrap.uncertainty_tumor` `O.nls.tumor_fit` ∂_t x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.time` | Time evolution of the state |
| `O.nls.tumor_fit` | O · nls · tumor fit operator |
| `S.bootstrap.uncertainty_tumor` | S · bootstrap · uncertainty tumor operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Cancer modeling |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | tumor_growth_parameter_vector |
| Noise model | lognormal |
| Integration axis | time |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 4. Measurement Model

Existence of the recovered tumor_growth_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 ~= 20); imaging_noise_percent dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Lognormal sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | volume_prediction_RMSE_percent |
| Secondary | time_to_progression_error_days |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Tumor volume mm3 | — | 1000 |
| N imaging timepoints | — | 8 |
| Imaging noise percent | — | 10 |
| Growth rate lambda day | — | 0.05 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Tumor volume mm3 | — | 10 – 100000 |
| N imaging timepoints | — | 3 – 50 |
| Imaging noise percent | — | 2 – 50 |
| Growth rate lambda day | — | 0.001 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 15 volume_prediction_RMSE_percent

| Metric | Range |
|---|---|
| Volume prediction rmse percent | 2 – 50 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **volume_prediction_RMSE_percent**, with κ = `500` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x620e8df375d9196c6a63965210a58ca98b17a9b9ce5db9fec21c68b1431358f6`
- **Chain tx hash:** `0x1778f33dfa3227c2f3e71c47c070508f97e6e04a38bcc50774df1a2470f4e149`
- **Chain block:** `41555219`

---

## File Mapping

This bundle consists of: `L1-410.md`, `L1-410.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-410.md` | Source of truth — edit this | Human or LLM |
| `L1-410.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 `L1-410`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._