Tissue laboratory · Cellular laboratory · Experimental curves & verification
Phase 2: material history in physical time
Both laboratories share a finite-strain continuum solver. Phase 2 adds three equilibrium material laws, a mixed macrocell pressure formulation and multiplicative finite-strain Maxwell branches. The original Release 01 elastic engine remains selectable as a numerical reference. Neither engine models puncture, neuronal injury or patient-specific tissue.
Material laws, stresses and units
Let F be the deformation gradient, J = det F, and λ̄ᵢ the volume-preserving principal stretches. The equilibrium strain-energy options are:
Ogden: W = 2μ/α² [Σ λ̄ᵢ^α − 3] Neo-Hookean: W = μ/2 (Ī₁ − 3) Mooney–Rivlin: W = μ/2 [(1−f)(Ī₁−3) + f(Ī₂−3)] Ī₁ = tr(J^(−2/3) F Fᵀ), Ī₂ = 1/2 [Ī₁² − tr(B̄²)]
Internal units are millimetres, kilopascals, seconds, millinewtons and microjoules. Thus 1 kPa × 1 mm² = 1 mN. Stress is derived from the energy; the displayed Green-strain norm is ||(FᵀF−I)/2|| and the stress overlay is von Mises Cauchy stress. Neither is an injury probability. Chart forces are signed nominal applied forces or reactions; the large force metric displays magnitude.
Nearly incompressible formulation
Six conforming linear tetrahedra share one parent-cell pressure. Deviatoric energy is fully integrated. An independent constant pressure is statically condensed, with reference-volume-weighted mean J̄ in each macrocell:
Πvol = Σcells Vc [p (J̄−1) − p²/(2K)] p = K (J̄−1) Πstabilization = 1/2 s μ₀ Σelements Ve (Je−J̄)² K = 2 μ₀ (1+ν) / [3(1−2ν)], μ₀ = μ∞ + Σ μbranch
The default stabilization s = 1 penalizes local volume fluctuations at the shear scale rather than imposing a large independent bulk penalty on every tetrahedron. A displacement-only tetrahedral comparator uses K(Je−1)²/2 instead. The reported maximum macrocell volume error and within-cell fluctuation make the two constraints visible.
This is a locking-resistant structured-mesh approximation, not an exactly incompressible formulation or a general stability proof for arbitrary tetrahedral meshes. Pressure is constant per parent cell; subcell pressure variation is not resolved. The nonrelaxing bulk modulus and selected Poisson ratio are modeling choices, not parameters measured by the cited fits. Mesh refinement is still necessary, particularly around implant corners and contact.
Finite-strain viscoelasticity
The published Ogden/Maxwell construction follows Budday et al. (2017), Viscoelastic parameter identification of human brain tissue, equations 18–23. Each branch uses the multiplicative decomposition F = FᵉFᵛ and a stored reference-frame inverse viscous metric. A backward-Euler logarithmic return mapping minimizes the branch energy plus η/Δt times the squared change in elastic principal logarithmic strain, subject to zero trace. Its derivative supplies the trial stress. The small-strain branch relaxation time is η/μ.
Trial evaluations never mutate material history. A new history is committed only after the global free-force residual converges. The next physical increment starts from that accepted state. Failed or canceled increments do not become completed time steps. The finite-Maxwell branch law retains the source Ogden exponent when a different equilibrium law is selected; that combination is explicitly a modified, uncalibrated comparison, not the published fit.
The physical-time protocols are ramp-and-hold stress relaxation, force-controlled creep, loading followed by force-free recovery, and one sinusoidal cycle. Recovery releases traction rather than forcing the tissue back to its initial shape. These are quasistatic material-history calculations without inertia, not dynamic wave propagation. Optimizer iterations and wall-clock rendering speed are not physical time.
Published presets versus refitted static curves
Eight presets transcribe Tables 4 and 6 of the cited viscoelastic study: cortex, basal ganglia, corona radiata and corpus callosum, each with first-cycle and third-cycle fits. These are averaged ex vivo regional fits. The white-matter models are isotropic; they do not introduce fiber-direction anisotropy. Source-reported loading-mode R² values are shown, including negative tensile R². Poor source fits are not concealed by solver convergence.
Twelve additional static presets are independently fitted to the pinned Living Matter Lab CANN experimental-curve release, jointly across tension, compression and shear. They do not infer viscosity from static data. The comparison page shows measurements, held-out points, model curves and errors. Curve-point and loading-mode holdouts do not constitute independent-donor validation. No confidence intervals are invented: the 0.75× and 1.25× material multipliers are sensitivity scenarios only.
Numerical verification and budgets
The numerical tests check force–energy gradients, frame invariance, pure dilation, the Ogden/neo-Hookean special case, finite-strain return mapping, history immutability, viscous metric positivity and unit determinant, dissipation, fixed and traction boundaries, cyclic sign reversal, scaling, rejection of unconverged history and exact affine anatomical advection.
A separately assembled Python/JAX energy with SciPy trust-region Newton independently verifies four nonlinear compression cases: mixed and full-volume formulations with neo-Hookean and Mooney–Rivlin laws. Displacement relative L₂ and reaction relative error must each be below 10⁻⁴. This is an independently implemented numerical reference, not a comparison against a commercial or mature FEM package. NumPy energy differences also check Ogden stress. Continuous-time SciPy DOP853 integration checks finite-Maxwell material history; reducing Δt from 0.5 to 0.25 to 0.125 seconds reduced the recorded peak-normalized stress error from about 2.11% to 1.08% to 0.55%.
The recorded compression sweep at ν = 0.499 changed reaction magnitude by about 3.2% relative to the 8-cell mixed result. The displacement-tetrahedron comparison changed about 80.3% relative to its 8-cell result (44.5% relative to its 4-cell result). This bounded result demonstrates reduced locking in that case, not mesh independence for all experiments. Refined meshes and stronger incompressibility can substantially exceed the interactive compute target.
The browser starts with an explicitly labeled 4-cell-per-axis interactive mesh. Solves use a 25-second elapsed computation budget; geometry loading is separate. A run that exceeds the budget stops at its last accepted state and is marked incomplete. The default history is 10 physical seconds with a 2-second ramp and maximum timestep of 0.5 seconds; the ramp receives at least four increments. Lower timesteps and finer meshes trade time for resolution, not an automatic guarantee of accuracy.
The normalized free-force residual is ||gfree||₂/(μ∞ L²). Standard and time-history tolerance is 10⁻⁵; the detailed static button uses 2×10⁻⁶. The local J ≤ 0.08 rejection is a numerical safeguard, not tissue damage. “Converged” denotes equilibrium of the discretized model, not experimental, spatial or temporal convergence.
Boundary conditions and contact limits
The lower sample face is clamped. Prescribed shear fixes all upper displacement components; stretch/compression prescribe vertical motion while allowing tangential motion. Force-controlled protocols free the upper face and apply reference-area-weighted nominal traction. Grabbing prescribes translation of a finite surface patch. The curved cap is a stress-free geometric option, not a pia membrane.
Static spherical indentation uses frictionless reference-area-weighted nodal penalty contact. Overlap is reported as numerical error. Static micromotion prescribes a bonded rigid 250 × 250 µm cuboid in an existing tract; changing depth rebuilds unloaded geometry, not a puncture history. Generic dimensions are not a Neuropixels specification. Contact, sharp-corner stresses, pia, friction, cutting and electrode bending remain separate validation tasks. Physical-time contact protocols are not exposed in this release.
Cellular anatomy and exports
The cellular page retains the attributed H01 proofread skeleton subset, described by Shapson-Coe et al. (2024). Credit: Lichtman Laboratory, Harvard University, and Connectomics at Google; CC BY 4.0 as recorded in the source manifest. An 80 µm crop retains native relative coordinates with translation only. Existing source simplification has a 0.7 µm positional tolerance.
Tubes interpolate measured skeleton radii and follow the continuum displacement through barycentric embedding. They are not dense segmented membranes, independently solved cells, or proof that a millimetre-scale constitutive fit is valid microscopically. Oversized soma-skeleton radii are omitted rather than portrayed as segmented soma membranes. No synthetic population fills missing tissue.
JSON exports retain parameters, units, mesh, the displayed field, accepted time-history positions and diagnostics, completion status, final accepted time, source provenance and the final inverse viscous metrics when the protocol completes. Replay can show an earlier accepted state; final metrics are explicitly labeled final. Incremental solver potential is distinct from stored material energy and dissipation. Exported state is an audit record; resuming an imported transient checkpoint is not implemented.
Import a segmented surface crop
Local surface JSON must already lie inside x,z ∈ [−40,40] µm and y ∈ [0,80] µm. Outside vertices are rejected, not rescaled or clamped. Limits are 100,000 vertices, 300,000 triangles and 15 MB. Source and license are mandatory but remain importer-supplied.
{
"schema": "bts-cellular-surface-v1",
"dataset": "dataset and exact region identifier",
"sourceUrl": "https://source.example/data",
"license": "the applicable data license",
"units": "um",
"positions": [0, 20, 0, 1, 20, 0, 0, 21, 0],
"indices": [0, 1, 2],
"note": "Segmentation version, source object IDs and coordinate transform"
} This triangle is a schema example, not biological data. Dense EM membranes, heterogeneous cellular mechanics, fluid transport, anisotropic tracts and implant puncture are not claimed by Phase 2.