"Virtual Element Method for oral biofilm mechanics: from confocal images to viscoelastic fracture on arbitrary polygonal meshes"
- Computational Mechanics (Wriggers editor, IKM ホーム) — 最有力
- Computer Methods in Applied Mechanics and Engineering (CMAME)
- International Journal for Numerical Methods in Engineering (IJNME)
- 世界初: VEM をバイオフィルム力学に適用
- Confocal → VEM 2-step pipeline: FEM の 5-step を大幅短縮
- DI-dependent 粘弾性 VEM: SLS + Simo 1987 on arbitrary polygons/polyhedra
- Phase-field 剥離 on VEM: DI → G_c マッピングによるバイオフィルム破壊
- Growth-coupled VE-VEM: 微生物動態 → 時間発展する粘弾性応答
- 口腔バイオフィルム: 構造化多菌種コミュニティ, dysbiosis → 歯周病
- 力学的性質 (E, G_c, η) が菌種組成に依存 (Pattem 2018/2021, Peterson 2015)
- 剥離 (detachment) = 界面破壊 → 力学モデルが必須
- 既存 FEM アプローチの限界: 構造メッシュ生成がボトルネック
- 任意多角形/多面体要素 (Beirão da Veiga 2013, 2014)
- Confocal コロニー = Voronoi セル = VEM 要素 (1:1 対応)
- Remesh 不要 → 成長・剥離・適応細分化に自然に対応
- IKM の VEM 固体力学 (Wriggers 2019, 2024; Aldakheel 2018)
- 12 モジュールの統合フレームワーク (Table 1)
- DI-dependent 構成則: E(DI), G_c(DI), SLS(DI)
- 2D/3D machine precision 検証
- Confocal → VEM パイプラインのデモ
- Virtual element space, elliptic projection Π^∇
- Stiffness decomposition: K = K_π + K_stab
- 2D: 6 poly basis (3 RBM + 3 strain), edge integrals
- 3D: 12 poly basis (3 trans + 3 rot + 6 strain), face integrals
- Stabilization: α_s tr(C)|E| (I - Π D)^T (I - Π D)
- Vertex + edge midpoint DOFs → 4n_v DOFs/element
- 12 poly basis: 3 RBM + 3 linear strain + 6 quadratic
- Analytical strain energy via sub-triangulation Gauss quadrature
- Volume correction for div(σ(p_α)) ≠ 0 in quadratic modes
- Standard Linear Solid: σ = C_inf ε + h(t)
- Simo exponential integrator: h_{n+1} = exp(-dt/τ) h_n + γ C_1 Δε
- Algorithmic tangent: C_alg = C_inf + γ C_1
- DI → SLS parameters: E_inf(DI), τ(DI), E_1(DI), η(DI)
- W(F) = μ/2(I₁-2) - μ ln(J) + λ/2(ln J)²
- Newton-Raphson + load stepping + line search
- DI → μ, λ via E(DI), ν
- Aldakheel 2018 framework: staggered u-d solve
- Spectral decomposition: ψ⁺/ψ⁻ tension-compression split
- DI → G_c(DI): commensal tough, dysbiotic fragile
- Irreversibility: d_{n+1} ≥ d_n
- Bilinear traction-separation law (Park-Paulino-Roesler 2009)
- Interface elements at tooth-biofilm boundary
- DI → σ_max, δ_c: mixed-mode coupling
- A posteriori error estimator: ZZ-type stress recovery
- Dörfler marking strategy
- Crack-tip indicator: η = w₁|∇d| + w₂ ψ⁺/G_c
- Field transfer (d, ψ history) via nearest-neighbor interpolation
- DI = -Σ φᵢ ln φᵢ / ln 5 (normalized Shannon entropy)
- Hamilton 5-species ODE → φᵢ(t) → DI(t)
- TMCMC calibration (20 params, 4 conditions)
- E(DI) = E_min + (E_max - E_min)(1-DI)^n
- E_max = 1000 Pa, E_min = 30 Pa, n = 2
- 文献根拠: Pattem 2018 (30× range), Gloag 2019, percolation theory (n=2)
- Mechanistic support (TMCMC paper, Section 2.4b):
- Composite model: φ_EPS × cross-link diversity → hydrogel scaling (de Gennes) → Mori-Tanaka
- 5-model BF comparison: DI decisive best (Bhatt=25.45), Composite 2nd (BF=2965)
- DI serves as proxy for φ_EPS × cross-link diversity → phenomenological law has first-principles basis
- G_c(DI) = G_c_min + (G_c_max - G_c_min)(1-DI)^n
- G_c_max = 0.5 J/m², G_c_min = 0.01 J/m²
- E_inf(DI), E_0 = 2 E_inf, E_1 = E_0 - E_inf
- τ(DI) = τ_min + (τ_max - τ_min)(1-DI)^n
- η = E_1 τ
confocal → voxel → marching cubes → tet mesh → Abaqus input
confocal → colony detection → Voronoi tessellation → VEM
- 5ch fluorescence → species classification
- Connected component → colony centroid
- Voronoi tessellation (scipy.spatial)
- Per-colony DI assignment → per-element material
- Demo: Heine 2025 FISH images
- Patch test: 10⁻¹⁸ (2D), 10⁻¹⁹ (3D)
- h-convergence: L² = 2.14, H¹ = 1.29 (Voronoi)
- VEM vs FEM comparison (Table)
- P₂ vs P₁: 15-45% stress accuracy improvement
- Laterally confined step strain — analytical solution
- 2D: 64 Voronoi cells, error = 1.3×10⁻¹⁵
- 3D: 27 hexahedral cells, error = 4.9×10⁻¹⁶
- Stress relaxation: monotonic decrease ✓
- Long-time limit: σ → E_inf ε/(1-ν²) ✓
- Instantaneous response: σ₀ = E_0 ε/(1-ν²) ✓
- Cook's membrane: Newton-Raphson convergence
- Linear vs nonlinear: 43% displacement difference at ε~1%
- Load stepping: 10 steps, ~25 NR iterations
- Single edge notch: crack path comparison
- DI-dependent: dysbiotic cracks first (G_c = 0.01 J/m²)
- Step 18: catastrophic failure (d: 0.33 → 1.0)
- Hamilton ODE → DI(t) → SLS(t) → VE-VEM
- 3 conditions: CS, DH, DS
- Key result: CS stiffens (DI↓, E↑, τ↑) vs DS softens (DI↑, E↓, τ↓)
- Fig: 3-condition stress relaxation comparison
- Fig: DI(t), E_inf(t), τ(t) evolution
- DI gradient: commensal periphery + dysbiotic center
- GCF pressure loading
- Dysbiotic center cracks → commensal survives
- Adaptive refinement at crack tip: 40 → 121 cells
- Fig: Phase-field evolution (d field snapshots)
- Fig: Load-displacement curve with failure point
- Bilinear TSL with DI-dependent σ_max
- Progressive debonding from weak (dysbiotic) center
- Load redistribution to intact (commensal) periphery
- Fig: Interface traction distribution at progressive load factors
- Heine 2025 FISH → Voronoi → VEM
- Per-colony DI → spatially resolved E, G_c
- von Mises stress field on confocal-derived mesh
- Fig: Confocal image → Voronoi mesh → stress field (side by side)
- Spatial DI variation → spatially varying SLS params
- Commensal (left) 59.7% relaxation vs dysbiotic (right) 75.0%
- Fig: Spatial stress field at t=0, t=τ, t=3τ
- Pipeline 短縮 (5-step → 2-step)
- Per-colony resolution without sub-element averaging
- Topology changes handled naturally
- Convergence rates competitive with FEM
- Simo 1987 × VEM: 文献に前例なし
- Machine precision → 実装の完全な正しさ
- Growth-coupled: 組成変化 → 力学応答を一気通貫
- Aldakheel 2018 をバイオフィルムに初適用
- DI → G_c: 組成から破壊靭性を予測
- Adaptive refinement: 計算コスト削減
- E(DI), G_c(DI), SLS(DI) は partially supported (直接同時計測データなし)
- 小変形仮定 (Neo-Hookean は大変形対応済み)
- 2D plane-stress (3D は検証済み、応用は future work)
- 粘弾性は SLS 1 branch のみ (Prony series 拡張可能)
- Confocal pipeline はシンセティック画像でデモ
- VEM × TMCMC integration (Bayesian inference with VEM forward model)
- 3D confocal z-stack → polyhedral VEM
- DeepONet surrogate on VEM meshes
- 実験検証: Sanz-Martin 2022 or 新規 confocal データ
- CLASI-FISH × U-Net によるセグメンテーション自動化（先行研究なし = novelty）
- 14ドナー, ~50k patches (256×256), Morillo-Lopez 2022 データ
- 先行 ML biofilm segmentation: Sadiq 2021 (Dice 0.90), Zhang 2022 (Dice 0.87), いずれも非 oral
- CLASI-FISH multi-species oral biofilm に ML 適用は世界初
- 3D Light Sheet Biofilm データ (Zenodo 10.5281/zenodo.18154035) で VEM 3D パイプラインを検証
- P.aeruginosa 単種: 282×339×714 voxels (189MB float32 TIFF)
- S.aureus + P.aeruginosa dual-species: 404×428×398 voxels (135MB float32 TIFF)
- CC-BY-4.0, open access, DL済み (
3d_data/) - 3D Voronoi VEM パイプライン実証済み (
pipeline_3d_real.py):- PA single: 65 Voronoi cells, E=[42,237] Pa, |u|=0.0023 µm, 11s total
- SAPA dual: 131 cells, DI=[0.18,0.79], E=[74,682] Pa, |u|=0.0005 µm, 37s
- Vertex cleanup (2196→352) で singular matrix 問題を解決
- VEM vs FEM tet benchmark (
benchmark_3d_vem_vs_tet.py):- VEM: 80 cells (1:1 colony correspondence), 1290 DOFs
- FEM: 482 tets, 282 DOFs
- VEM advantage: 1 colony = 1 element → per-colony DI/E 直接割当て
- Phase-field on real geometry (
phase_field_real_3d.py):- SAPA 2D cross-section → 69 VEM elements, DI-dependent G_c
- PA-dominant 領域 (G_c=0.038) から優先的にクラック進展
- 30 steps で d_max=0.85 (progressive damage)
- Mark Welch (MBL) に 3D oral z-stack 共有を打診 → future collaboration
- CLASI-FISH hedgehog structures (10+ taxa, PNAS 2016)
- Raw z-stacks は未公開 → 共有打診が最も生産的なパス
- 公開 3D oral biofilm データセットは存在しない (2025年時点)
- 取得済み (Zenodo 10.5281/zenodo.18154035, CC-BY-4.0):
PA_cluster2_3d.tif: P.aeruginosa 282×339×714, light sheet 3D再構成SAPA_cluster2_3d.tif: S.aureus+P.aeruginosa 404×428×398, dual-species
- HiPR-FISH (Shi et al. 2020): 100+ taxa, probe tools on GitHub, raw images 要著者連絡
- Mark Welch (MBL): CLASI-FISH hedgehog oral z-stacks, 要著者連絡
| 著者 | 年 | 手法 | 対象 | Dice/Acc | 公開 |
|---|---|---|---|---|---|
| Sadiq et al. | 2021 | U-Net | P.aeruginosa CLSM | ~0.90 | No |
| Zhang et al. | 2022 | Mask R-CNN + U-Net | CLSM biofilm | 0.87 | Partial |
| Jeckel et al. | 2023 | CNN (BiofilmQ) | Phase+蛍光 | tracking | Yes |
| Kim et al. | 2022 | ResNet-50 | Confocal 4分類 | 94.2% | No |
| Rani et al. | 2023 | U-Net + attention | 廃水 biofilm | 0.91 | No |
| Ours (proposed) | 2026 | U-Net | CLASI-FISH oral 5+ species | TBD | Yes |
- 世界初: VEM × バイオフィルム力学
- 12 ソルバー, 120+ テスト, machine precision
- DI-dependent 弾性/粘弾性/破壊の統一フレームワーク
- Confocal → VEM 2-step pipeline で実験データ直結
- 3D real biofilm 実証: light sheet TIFF → 131 Voronoi cells → VEM 応力解析 (37s)
- Phase-field on real geometry: PA-dominant 領域から優先的にクラック進展
- IKM の VEM 固体力学 + TMCMC 微生物動態 = 新しい計算バイオメカニクス
| Fig | Content | Source |
|---|---|---|
| 1 | Pipeline comparison: FEM 5-step vs VEM 2-step | New schematic |
| 2 | VEM schematic: projection Π^∇, polygon element | New schematic |
| 3 | Constitutive laws: E(DI), G_c(DI), SLS(DI) + literature overlay | generate_grand_showcase.py panel (a) extended |
| 4 | h-convergence: VEM vs FEM, L²/H¹ rates | vem_convergence_study.py |
| 5 | VE-VEM validation: confined relaxation, 2D & 3D analytical match | vem_viscoelastic.py + vem_3d_viscoelastic.py |
| 6 | P₁ vs P₂: stress accuracy comparison | vem_p2_elasticity.py |
| 7 | Neo-Hookean: linear vs nonlinear displacement | vem_nonlinear.py |
| 8 | Phase-field evolution: d-field snapshots + load-displacement | vem_phase_field.py |
| 9 | Adaptive refinement: mesh evolution at crack tip | vem_adaptive_fracture.py |
| 10 | CZM debonding: interface traction at progressive loads | vem_czm.py |
| 11 | Growth-coupled VE-VEM: 3-condition DI(t), σ(t), E(t) evolution | vem_viscoelastic_growth.py |
| 12 | DI gradient + viscoelasticity: spatial stress at t=0, τ, 3τ | vem_viscoelastic.py demo |
| 13 | Confocal → VEM: FISH image → Voronoi → stress field | vem_confocal_pipeline.py |
| 14 | Grand showcase: 8-panel overview (graphical abstract candidate) | generate_grand_showcase.py |
| 15 | 3D real biofilm: PA/SAPA overview (colony, DI, E, | u |
| 16 | VEM vs FEM tet benchmark: mesh comparison table + displacement | benchmark_3d_vem_vs_tet.py |
| 17 | Phase-field on real geometry: DI → G_c → progressive damage | phase_field_real_3d.py |
| 18 | Single vs dual comparison: DI/E/ | u |
- Beirão da Veiga et al. (2013, 2014) — M3AS
- Sutton (2017) — Numer. Algorithms
- Ahmad et al. (2013) — Comput. Math. Appl.
- Wriggers, Hudobivnik (2019) — Comput. Mech.
- Wriggers, Aldakheel, Hudobivnik (2024) — Springer book
- Aldakheel et al. (2018) — CMAME
- Nguyen-Thanh et al. (2018) — CMAME
- Artioli et al. (2017) — P₂ VEM
- Simo (1987) — CMAME — exponential integrator
- Simo, Hughes (1998) — Computational Inelasticity
- Xu, Junker, Wriggers (2025) — Space-time VEM (if published)
- Bourdin, Francfort, Marigo (2000, 2008)
- Miehe, Hofacker, Welschinger (2010) — CMAME
- Ambati, Gerasimov, De Lorenzis (2015) — CM
- Park, Paulino, Roesler (2009) — IJNME
- Xu, Needleman (1994)
- Pattem et al. (2018) — Sci Rep — AFM + 16S
- Pattem et al. (2021) — Sci Rep — hydrated biofilm
- Gloag et al. (2019) — J Bacteriol
- Peterson, Stoodley (2015) — FEMS Microbiol Rev
- Flemming, Wingender (2010) — Nat Rev Microbiol
- Klempt et al. (2024) — staggered coupling
- Heine et al. (2025) — 5-species FISH
- Sahimi (1994) — Applications of Percolation Theory
- Arbabi, Sahimi (1993) — Phys Rev B
| Week | Task |
|---|---|
| 1 | Sec 2 (数学) + Sec 5 (検証) — 既存コード結果をまとめる |
| 2 | Sec 3 (構成則) + Sec 4 (pipeline) — 図作成 |
| 3 | Sec 6 (応用) — growth-coupled, phase-field, confocal 結果 |
| 4 | Sec 1 (intro) + Sec 7 (discussion) + Sec 8 (conclusion) |
| 5 | 全体推敲 + Wriggers/Aldakheel 先生にドラフト共有 |
- Nishioka K., Aldakheel F., Wriggers P.
- IKM, Leibniz Universität Hannover