# ⚛  L1 Principle — Windkessel Cardiac Afterload Model

**ID:** `L1-401` · **Status:** ⊙ Testnet (genesis catalog)

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

---

## 🧠 1. Introduction

**Windkessel Cardiac Afterload Model** is a **parameter-estimation problem** whose unknown lives in **windkessel parameter vector** space, within the **Cardiovascular physiology** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **aortic pressure flow measurement** sensing mechanism.

The forward operator applies, in order: time evolution of the state; O · least squares · pressure fit operator; S · gradient · ode params operator.

Observations are corrupted by measurement gaussian. Existence of the recovered windkessel_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); nonlinearity_pressure_compliance 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** → Time derivative → O · least squares · pressure fit → S · gradient · ode params → **y** (detector).

```
y = `S.gradient.ode_params` `O.least_squares.pressure_fit` ∂_t x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.time` | Time evolution of the state |
| `O.least_squares.pressure_fit` | O · least squares · pressure fit operator |
| `S.gradient.ode_params` | S · gradient · ode params operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Cardiovascular physiology |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | windkessel_parameter_vector |
| Noise model | measurement_gaussian |
| Integration axis | cardiac_cycle |
| Difficulty delta | 3 |
| L dag | 2.2 |

## 📡 4. Measurement Model

Existence of the recovered windkessel_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); nonlinearity_pressure_compliance dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Measurement gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | pressure_waveform_RMSE_mmHg |
| Secondary | parameter_identifiability_index |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Heart rate bpm | — | 70 |
| N cardiac cycles | — | 10 |
| Systolic pressure mmhg | — | 120 |
| Diastolic pressure mmhg | — | 80 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Heart rate bpm | — | 40 – 120 |
| N cardiac cycles | — | 3 – 100 |
| Systolic pressure mmhg | — | 80 – 200 |
| Diastolic pressure mmhg | — | 50 – 100 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 2.0 pressure_waveform_RMSE_mmHg

| Metric | Range |
|---|---|
| Pressure waveform rmse mmhg | 0.2 – 10.0 |

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x9e264bd4cb29c0c0d2805ff69313eb2d909d79a18da8bfea36b01e6e83ce6b2c`
- **Chain tx hash:** `0xbb2f99b0073c1b641d5f7611f14337dc55d81de4da2937f931619f6e60ed9109`
- **Chain block:** `41555217`

---

## File Mapping

This bundle consists of: `L1-401.md`, `L1-401.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-401.md` | Source of truth — edit this | Human or LLM |
| `L1-401.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 `L1-401`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._