# ⚛  L1 Principle — Hemodynamics (Navier-Stokes Cardiovascular)

**ID:** `L1-400` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Cardiovascular modeling*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** 3D velocity pressure field
> **📡 Carrier:** N/A · **🌫 Noise:** measurement gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555217

---

## 🧠 1. Introduction

**Hemodynamics (Navier-Stokes Cardiovascular)** is a **parameter-estimation problem** whose unknown lives in **3D velocity pressure field** space, within the **Cardiovascular modeling** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **blood flow velocity measurement** sensing mechanism.

The forward operator applies, in order: gradient / divergence with respect to position; S · fsi · fluid structure operator; O · chi2 · 4dflow mri operator.

Observations are corrupted by measurement gaussian. Existence of the recovered 3D_velocity_pressure_field 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 ~= 2000); MRI_velocity_noise dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Measurement gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Spatial derivative → S · fsi · fluid structure → O · chi2 · 4dflow mri → **y** (detector).

```
y = `O.chi2.4dflow_mri` `S.fsi.fluid_structure` ∇ x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.space` | Gradient / divergence with respect to position |
| `S.fsi.fluid_structure` | S · fsi · fluid structure operator |
| `O.chi2.4dflow_mri` | O · chi2 · 4dflow mri operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Cardiovascular modeling |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | 3D_velocity_pressure_field |
| Noise model | measurement_gaussian |
| Integration axis | 3d_vascular |
| Difficulty delta | 5 |
| L dag | 4.5 |

## 📡 4. Measurement Model

Existence of the recovered 3D_velocity_pressure_field 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 ~= 2000); MRI_velocity_noise dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Measurement gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | WSS_RMSE_Pa |
| Secondary | pressure_drop_RMSE_mmHg |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Re reynolds | — | 2000 |
| Mri voxel mm | mm | 2 |
| N mesh elements | — | 500000 |
| Cardiac cycle s | s | 1 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Re reynolds | — | 100 – 5000 |
| Mri voxel mm | mm | 0.5 – 5.0 |
| N mesh elements | — | 10000 – 5000000 |
| Cardiac cycle s | s | 0.5 – 2.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.5 WSS_RMSE_Pa

| Metric | Range |
|---|---|
| Wss rmse pa | 0.05 – 3.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **WSS_RMSE_Pa**, with κ = `100000.0` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x54810c5dfcd075e620d4cd14e8c1bb3861660ebbc04b51b54b265d32bc9178c8`
- **Chain tx hash:** `0xbd399fe67a4e1d72025a46ec52ba758d45d1ac6c7ded7d2536a4111985f8a6fa`
- **Chain block:** `41555217`

---

## File Mapping

This bundle consists of: `L1-400.md`, `L1-400.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-400.md` | Source of truth — edit this | Human or LLM |
| `L1-400.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 `L1-400`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._