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Cardiac 4D-flow MRI with Hemodynamic Biomechanics

Medical Imaging · Cardiovascular blood-flow and pressure recovery via MR phase-contrast + Navier-Stokes coupling (multi-physics joint inverse)

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⬇ L1-510.md

# ⚛ L1 Principle — Cardiac 4D-flow MRI with Hemodynamic Biomechanics **ID:** `L1-510` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Cardiovascular blood-flow and pressure recovery via MR phase-contrast + Navier-Stokes coupling (multi-physics joint inverse)* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 4D velocity pressure joint > **📡 Carrier:** radio_wave_with_blood_flow · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41552305 --- ## 🧠 1. Introduction **Cardiac 4D-flow MRI with Hemodynamic Biomechanics** is a **nonlinear inverse problem** whose unknown lives in **4D velocity pressure joint** space, within the **Cardiovascular blood-flow and pressure recovery via MR phase-contrast + Navier-Stokes coupling (multi-physics joint inverse)** sub-domain of **Medical Imaging**. Measurements consist of radio wave with blood flow via a **phase contrast mri with navier stokes** sensing mechanism. The forward operator applies, in order: Bloch-equation tip of the magnetisation vector; L · flow encoding gradient operator; L · gradient echo operator; L · larmor precession operator; L · phase extract operator; L · navier stokes constraint operator; L · boundary conditions operator; int · spatial temporal operator. Observations are corrupted by additive Gaussian noise. Existence of joint (v(r, t), p(r, t)) is guaranteed within the declared Omega bounds and chamber geometry. Uniqueness holds under physiologically-consistent boundary conditions (no-slip walls, prescribed inflow/outflow, periodic-cardiac-cycle) and adequate venc/SNR; pressure recovery is conditionally unique up to a constant (Helmholtz-Hodge decomposition) — typical practice fixes pressure at one anatomical reference. Stability is moderately conditioned (kappa_eff ~ 30 after NS-regularized variational reconstruction) — venc_aliasing dominates aliasing-induced velocity errors; partial_volume_at_walls dominates wall-shear-stress bias; eddy_current_offset contributes a systematic background-velocity bias; viscosity_uncertainty contributes a scaling factor for shear-related quantities. Joint Hadamard well-posedness for the coupled MR-PC + NS forward is established by Markl et al. 2012 (foundational 4D-flow review), Stankovic et al. 2014 (clinical 4D-flow), Bertoglio-Caiazzo 2015 (NS-regularized 4D-flow inverse), Pereira et al. 2016 (assimilation), Garcia et al. 2018 (NS-regularized clinical 4D-flow), and Soulat et al. 2020 (4D-flow benchmarking). ## ⚙ 2. Forward Model Physical chain: **x** → RF excitation pulse → L · flow encoding gradient → L · gradient echo → L · larmor precession → L · phase extract → L · navier stokes constraint → L · boundary conditions → int · spatial temporal → **y** (detector). ``` y = `int.spatial_temporal` `L.boundary_conditions` `L.navier_stokes_constraint` `L.phase_extract` `L.larmor_precession` `L.gradient_echo` `L.flow_encoding_gradient` B₁(t) x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.rf_excitation` | Bloch-equation tip of the magnetisation vector | | `L.flow_encoding_gradient` | L · flow encoding gradient operator | | `L.gradient_echo` | L · gradient echo operator | | `L.larmor_precession` | L · larmor precession operator | | `L.phase_extract` | L · phase extract operator | | `L.navier_stokes_constraint` | L · navier stokes constraint operator | | `L.boundary_conditions` | L · boundary conditions operator | | `int.spatial_temporal` | Int · spatial temporal operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | Cardiovascular blood-flow and pressure recovery via MR phase-contrast + Navier-Stokes coupling (multi-physics joint inverse) | | Carrier | radio_wave_with_blood_flow | | Problem class | nonlinear_inverse | | Solution space | 4D_velocity_pressure_joint | | Noise model | gaussian | | Integration axis | spatial_temporal | | Difficulty delta | 5 | | L dag | 8.3 | ## 📡 4. Measurement Model Existence of joint (v(r, t), p(r, t)) is guaranteed within the declared Omega bounds and chamber geometry. Uniqueness holds under physiologically-consistent boundary conditions (no-slip walls, prescribed inflow/outflow, periodic-cardiac-cycle) and adequate venc/SNR; pressure recovery is conditionally unique up to a constant (Helmholtz-Hodge decomposition) — typical practice fixes pressure at one anatomical reference. Stability is moderately conditioned (kappa_eff ~ 30 after NS-regularized variational reconstruction) — venc_aliasing dominates aliasing-induced velocity errors; partial_volume_at_walls dominates wall-shear-stress bias; eddy_current_offset contributes a systematic background-velocity bias; viscosity_uncertainty contributes a scaling factor for shear-related quantities. Joint Hadamard well-posedness for the coupled MR-PC + NS forward is established by Markl et al. 2012 (foundational 4D-flow review), Stankovic et al. 2014 (clinical 4D-flow), Bertoglio-Caiazzo 2015 (NS-regularized 4D-flow inverse), Pereira et al. 2016 (assimilation), Garcia et al. 2018 (NS-regularized clinical 4D-flow), and Soulat et al. 2020 (4D-flow benchmarking). | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | RMSE_per_velocity_component | ## 📏 5. Operating Range (Ω) **Center problem class:** `cardiac_4dflow_ns_regularized` · **Forward operator:** `cardiac_4dflow_joint_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 192 | | W | px | 192 | | Z | — | 64 | | Te ms | ms | 2.5 | | Tr ms | ms | 5 | | Snr db | dB | 24 | | Venc mps | — | 1.5 | | B0 field t | — | 3 | | Venc aliasing | — | 0 | | Voxel size mm | mm | 2.5 | | N cardiac phases | — | 25 | | Breath hold motion | — | 0 | | Eddy current offset | — | 0 | | Viscosity uncertainty | — | 0 | | Partial volume at walls | — | 0 | | Segmentation error at chambers | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 128 – 512 | | W | px | 128 – 512 | | Z | — | 32 – 256 | | Te ms | ms | 1.5 – 5.0 | | Tr ms | ms | 3.0 – 10.0 | | Snr db | dB | 5.0 – 38.0 | | Venc mps | — | 0.5 – 6.0 | | B0 field t | — | 1.5 – 7.0 | | Venc aliasing | — | 0.0 – 0.3 | | Voxel size mm | mm | 1.0 – 4.0 | | N cardiac phases | — | 10 – 50 | | Breath hold motion | — | 0.0 – 0.3 | | Eddy current offset | — | 0.0 – 0.2 | | Viscosity uncertainty | — | 0.0 – 0.3 | | Partial volume at walls | — | 0.0 – 0.4 | | Segmentation error at chambers | — | 0.0 – 0.3 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 26.0 | Metric | Range | |---|---| | Psnr db | 10.0 – 42.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `220` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xb25b2551bc781001bdbc0c456066a6dbda2ec0862615e8c6d13eaccd12c31009` - **Chain tx hash:** `0x6eef5f3c58e7f98baff189b5095bd56113d47f055cf3e3ebc3265addd1b22f58` - **Chain block:** `41552305` --- ## File Mapping This bundle consists of: `L1-510.md`, `L1-510.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-510.md` | Source of truth — edit this | Human or LLM | | `L1-510.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 `L1-510`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-510.md).
Regenerate L1-510.json so every field matches the Markdown.
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L2-510-001 Cardiac 4D-flow MRI with Hemodynamic Biomechanics — Spec Skeleton

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📋 JSON metadata

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    "regime": "Existence of joint (v(r, t), p(r, t)) is guaranteed within the declared Omega bounds and chamber geometry. Uniqueness holds under physiologically-consistent boundary conditions (no-slip walls, prescribed inflow/outflow, periodic-cardiac-cycle) and adequate venc/SNR; pressure recovery is conditionally unique up to a constant (Helmholtz-Hodge decomposition) \u2014 typical practice fixes pressure at one anatomical reference. Stability is moderately conditioned (kappa_eff ~ 30 after NS-regularized variational reconstruction) \u2014 venc_aliasing dominates aliasing-induced velocity errors; partial_volume_at_walls dominates wall-shear-stress bias; eddy_current_offset contributes a systematic background-velocity bias; viscosity_uncertainty contributes a scaling factor for shear-related quantities. Joint Hadamard well-posedness for the coupled MR-PC + NS forward is established by Markl et al. 2012 (foundational 4D-flow review), Stankovic et al. 2014 (clinical 4D-flow), Bertoglio-Caiazzo 2015 (NS-regularized 4D-flow inverse), Pereira et al. 2016 (assimilation), Garcia et al. 2018 (NS-regularized clinical 4D-flow), and Soulat et al. 2020 (4D-flow benchmarking).",
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  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Cardiovascular blood-flow and pressure recovery via MR phase-contrast + Navier-Stokes coupling (multi-physics joint inverse)",
  "title": "Cardiac 4D-flow MRI with Hemodynamic Biomechanics"
}

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