# Stabilized cell-surface force reconstruction: full-series technical report Date: 2026-07-20 Dataset: Sixt lightsheet, 30 timepoints (T000--T029) Status: final ## Executive conclusion The earlier project state had **not** reconciled the large matrix deformations with the very small surface-regularized forces. The main numerical problem was that SAENOpy's `reg_mask` had been interpreted as a hard zero-force mask. It is actually a regularisation mask: nodes outside it are unpenalized while fitting and are zeroed only after the inverse solve. The old output could therefore omit bulk forces that had helped fit the measured deformation. The new reconstruction fixes that accounting problem and produces a stable, equilibrium-compatible, surface-localized force field for every timepoint. At the selected common parameter, alpha = 1e7, the near-surface target displacement has a median frame-wise p90 of 0.402 um (maximum frame-wise p90 0.753 um; largest individual vector 1.408 um), while the total surface force has a median of 2.37 nN (range 0.514--5.865 nN). The fitted field retains the coherent large deformation: across frames the median fitted-vector cosine is 0.973 and the median fitted normal-displacement correlation is 0.910. Thus the low-nN branch is not obtained simply by ignoring the observed 0.5-um-scale motion. The most important hyperparameter result is positive. Whole-series contractility has a strict decade-wide plateau from alpha = 1e7 to 1e8: total force changes by a factor of only 1.152, pulling force by 1.118, the median absolute log-log force slope is 0.044 (p90 0.125), the median residual changes by only 1.010, and every solution in the interval passes the numerical quality gates. Alpha = 1e7 is the lowest-regularization edge of that plateau and is the recommended common value. Fifteen of 30 individual frames pass the same very strict decade-wide gate; the aggregate plateau is more stable because low-signal frames retain greater parameter sensitivity. The corrected cell-on-matrix traction also agrees with the requested biological pattern. Pulling is enriched at one or both cell ends in 29/30 frames (median end enrichment 1.320). Central pushing is enriched in 18/30 frames, and the combined end-pull/central-push pattern occurs in those same 18 frames. The dominant pull is at the front in 16 frames and at the rear in 14 frames. This supports a general end-pulling plus intermittent central pushing interpretation, not a claim that the pattern occurs in every frame. The bright internal segmentation has not been independently established as the nucleus, so the conservative description is "central/internal-signal-associated pushing." A supplemental surface-graph smoothness prior (weight 100, 5-um radius) was tested over the full series because it improved two representative spatial holdouts. It is **not** recommended as the primary quantitative reconstruction: the full-series quality-qualified plateau fails, only 7/30 frames pass the per-frame gate, and alpha = 1e7 passes all equilibrium/localization gates in only 73.3% of frames. The unsmoothed stabilized reconstruction is therefore the final result and the graph-smoothed screen is retained as a sensitivity test. ## Data and preserved inputs The analysis starts from the preserved 30-solver SAENOpy archive: `outputs/saenopy_files/current_runs/optimal_field_T013x050_T016x050_T000_to_T029_mesh3um_cell_traction.saenopy` For each timepoint, the archived tetrahedral mesh, target displacement, target mask, original broad regularisation domain, and solved displacement were extracted without resampling. The image-to-mesh geometry was reconstructed from the corresponding cell scene in: `outputs/viewer_runs/optimal_T013x050_T016x050_alpha5e11/timepoints/T###/scene.glb` The source voxel spacing is 0.145 um in x, y, and z. Depending on timepoint, the meshes contain 12,167--15,341 nodes and 63,888--81,312 tetrahedra. The cell shell contains 263--327 surface nodes; the union force domain contains 3,456--4,884 nodes. A geometry audit reproduced the archived broad regularisation and target masks for all 30 timepoints. ## Force sign and biological geometry SAENOpy stores the internal nodal reaction in `solver.mesh.forces`, while its GUI displays the equal-and-opposite cell-on-matrix traction. All biological metrics and new GLB arrows therefore use: `cell_traction = -solver.mesh.forces` With outward-oriented cell-surface normals, a negative normal component means pulling toward the cell and a positive component means pushing away. Earlier custom signed GLBs and pull/push labels used the opposite sign and are stale. The force magnitudes in those old files are not affected by this sign issue. The archived solver origin is the bounding-box center of the union of the cell surface and internal signal and can lie about 5 um from the actual cell-surface center. Biological regions therefore use the area-weighted surface center and the area-weighted long-axis principal component. The long-axis sign is oriented by the smoothed surface-center trajectory. Front and rear are the outer 20% area quantiles of this axis; the center is the middle third. Surface normals are explicitly reoriented about the area-weighted surface center. ## Stabilized inverse method The production reconstruction uses the following steps. 1. Keep the complete broad force domain in the inverse objective, so no fitted bulk force can disappear after the solve. 2. Define a 3-um shell around the segmented cell surface as the low-penalty force region. 3. Use fixed two-level quadratic regularisation rather than an iteratively updated Huber weight that can let isolated nodes escape. 4. Continue the bulk/surface penalty ratio through 1, 1e3, and 1e6. This moves support to the cell surface gradually and avoids the ill-conditioning caused by applying the full contrast from a zero initialization. 5. Add low-rank penalties for net force and radius-normalized net torque, both with weight 1.0. 6. Anchor on strong regularisation and follow a descending alpha continuation: 1e8, 5e7, 2e7, 1e7, 5e6, 2e6, and 1e6. 7. Reject or flag any solution with more than 2% bulk-force leakage, more than 5% normalized net force or torque, more than 10% of force on the top 1% of nodes, or maximum/p95 force greater than 10. The nonlinear material parameters inherited from the existing 1.2-mg/ml gel model are `k = 1449 Pa`, `d0 = 0.0022`, `lambda_s = 0.032`, and `d_s = 0.055`. The inverse step size is 0.33, with 40 maximum iterations, 12 minimum iterations, and relative convergence threshold 0.005. These parameters are now explicit in the command line and output metadata. ## Alpha selection and plateau test The screen contains 210 reconstructions: 30 frames times seven alpha values. All 210 completed and passed the basic per-solution quality checks. A plateau is not declared from force magnitude alone. A contiguous interval must contain at least three alpha values, span at least one decade, and satisfy all of: Here "overall contractility" is operationally evaluated by both the integrated surface-force magnitude and the integrated inward surface-normal pulling force. This avoids using a signed net force, which should approach zero for a balanced cell, while still requiring the explicitly contractile component to plateau. - total force max/min <= 1.25; - pulling force max/min <= 1.35; - median and p90 absolute log-log force slopes <= 0.10 and 0.25; - residual p90 max/min <= 1.10; - adjacent force-field cosine >= 0.90; - adjacent signed-traction correlation >= 0.80; - bulk leakage <= 2%, force/torque imbalance <= 5%; and - no force-concentration quality flag. The whole-series aggregate passes on [1e7, 1e8]. Its total-force max/min ratio is 1.1517, pulling-force ratio 1.1177, median slope 0.0439, p90 slope 0.1254, residual ratio 1.0096, and minimum quality-pass fraction 1.0. Alpha = 1e7 is inside the maximum-coverage plateau for 15/30 frames and is the recommended common value. It preserves more deformation information than choosing a higher point within the same plateau. The frame median total force falls from 6.91 nN at alpha = 1e6 to 2.37 nN at 1e7, 1.93 nN at 2e7, 1.84 nN at 5e7, and 1.79 nN at 1e8. This makes the transition into the high-alpha asymptote visible and prevents selecting an apparently convenient value below the plateau. ## Deformation fit and the large-displacement question At alpha = 1e7, the frame-wise target-displacement p90 ranges from 0.072 to 0.753 um and has median 0.402 um. The largest individual near-surface target vector is 1.408 um. The fit-residual p90 has median 0.142 um, or 0.370 of target p90; its p10--p90 range is 0.046--0.197 um. The median fitted-vector cosine is 0.973 (p10--p90 across frames 0.934--0.986), and the median fitted normal-field correlation is 0.910 (p10--p90 0.821--0.938). Total force and target p90 covary across frames with Pearson r = 0.728. These figures reconcile the apparent contradiction in the sense relevant to the inverse problem: nN-scale, surface-localized, balanced tractions reproduce most of the coherent deformation, including frames with p90 displacement above 0.5 um. They do **not** prove an absolute force calibration independent of the material law and optical-flow target. Large isolated vectors can also be registration outliers; they should not force a higher traction unless they are spatially coherent and independently reproducible. ## Nonlinear branch and predictive validation The inverse problem is non-convex. Ascending and descending alpha continuations on T000, T007, and T013 reach different solutions at alpha = 1e7. The ascending branch has 4.22--7.03 times more force with only a small residual improvement; the descending branch has the lower reported regularized objective in all three frames. For example, T007 gives 1.562 versus 10.976 nN, residual p90 0.0945 versus 0.0936 um, and force-field cosine 0.691. The canonical rule is therefore to start at alpha = 1e8 and follow the lower-objective descending branch, not to initialize directly at the desired alpha. A true spatial-block holdout removed one of five 18-um blocks from the inverse fit and predicted it afterward. At alpha = 1e7: | frame and branch | all-block R2 | near-surface R2 | normal r | normal sign agreement | median vector cosine | | --- | ---: | ---: | ---: | ---: | ---: | | T000 descending | 0.795 | 0.557 | 0.933 | 0.855 | 0.899 | | T000 ascending | 0.735 | 0.580 | 0.928 | 0.852 | 0.868 | | T007 descending | 0.322 | 0.436 | 0.819 | 0.881 | 0.894 | | T007 ascending | -0.328 | 0.068 | 0.749 | 0.874 | 0.886 | The low-force descending branch is therefore predictive, especially on T007; it is not merely an overregularized in-sample solution. A degree-normalized surface graph prior with weight 100 and 5-um neighbor radius improved the same holdouts further: T000 all/near R2 becomes 0.861/0.663 and T007 becomes 0.689/0.728. That representative-frame improvement does not generalize cleanly enough to justify the prior as the production estimator. In the complete 30-frame, four-alpha screen, the aggregate total-force and pull ratios over [1e7, 1e8] are numerically small (1.172 and 1.157), but the minimum per-alpha quality-pass fraction is only 0.733, so the quality-qualified plateau fails. Twenty of 120 solutions carry at least one flag: 18 torque-imbalance, six net-force-imbalance, and six bulk-leakage flags, with some solutions carrying multiple flags. At alpha = 1e7, eight frames fail, predominantly on torque balance. Only 7/30 frames pass the complete per-frame plateau gate. The median force is 0.458 nN at alpha = 1e7 and 0.301 nN at 1e8, while the median fit-residual p90 is 0.158 um, slightly worse than the 0.142-um reference result. The time-series force-versus-target-p90 correlation also falls from 0.728 to 0.630. The smooth screen still recovers end-pull enrichment in 26/30 frames and central push enrichment in 20/30, showing that the macroscopic localization is fairly robust. However, it should be used only as a labeled sensitivity/denoising view on quality-passing frames, not for absolute force scale or as the default arrow set. The production GLBs therefore use the unsmoothed stabilized alpha = 1e7 field. ## Cell behavior At alpha = 1e7: - end pulling is enriched in 29/30 frames; median enrichment is 1.320 and the p10--p90 interval is 1.057--1.534; - central pushing is enriched in 18/30 frames; median enrichment is 1.039 and the p10--p90 interval is 0.745--1.298; - the combined expected pattern occurs in 18/30 frames; - the front is the dominant pulling pole in 16 frames and the rear in 14; and - median direct local force/deformation sign agreement is 0.633. The direct pointwise force-versus-displacement correlation is only moderate (median 0.391; pooled 0.339). This is expected in a nonlocal elastic medium: a surface force affects displacement throughout the mesh, so displacement at a node is not determined only by the nearest traction arrow. The more appropriate forward-fit checks are strong (median normal-field correlation 0.910 and vector cosine 0.973), and spatial holdout confirms genuine predictive structure. The time series suggests a transition from predominantly rear-end pulling in T000--T012 to predominantly front-end pulling through much of T013--T26, with rear dominance returning at T027--T028. Central pushing is intermittent rather than universal. This is consistent with a cell alternately anchoring and pulling at an end while generating central/internal pressure during some constriction events. Establishing that the internal signal is specifically the nucleus would require an independent label or segmentation audit. ## Limitations and interpretation boundary 1. Absolute nN values remain conditional on the inherited nonlinear material parameters; the gel was not independently calibrated in this analysis. 2. The target displacement and its relaxed reference are inherited from the preserved archive. Registration uncertainty is not yet propagated into a formal force confidence interval. 3. The segmentation defines where surface forces are encouraged. A 3-um shell is appropriate for the current 3-um mesh but is not a universal biological length scale. 4. Non-convex branch ambiguity is controlled by a reproducible strong-alpha anchor, objective comparison, and holdout, but it is not mathematically eliminated. 5. Per-frame alpha plateaus occur in 15/30 frames. The robust conclusion is a whole-series contractility plateau and a common alpha, not that every weak frame is independently identifiable over a full decade. 6. Surface traction is projected conservatively with a 4-um kernel for quantitative regional metrics. The GLB display uses a smoother 8-um kernel for readability; arrows themselves retain the corrected cell-traction vectors. ## Reproducibility and outputs Primary reference screen: `outputs/saenopy_surface_stabilized_full_series_v1` Full biological validation: `outputs/saenopy_surface_stabilized_full_series_v1/behavior_validation` Corrected alpha = 1e7 force scenes: `outputs/viewer_runs/stabilized_surface_full_series_alpha1e07` Branch hysteresis validation: `outputs/saenopy_surface_stabilized_hysteresis_validation_v1` Spatial holdout validations: `outputs/saenopy_surface_stabilized_holdout_branch_T000_desc_v1` `outputs/saenopy_surface_stabilized_holdout_branch_T007_desc_v1` `outputs/saenopy_surface_stabilized_holdout_branch_T000_desc_smooth_w100_v1` `outputs/saenopy_surface_stabilized_holdout_branch_T007_desc_smooth_w100_v1` Rejected full-series graph-smoothness sensitivity screen: `outputs/saenopy_surface_stabilized_smooth_w100_full_series_v1` The implementation is covered by 23 passing tests, including mask extraction, force sign, geometry, surface projection, continuation, quality gates, and the surface graph operator.