Divergence of detachment forces in the finite Voronoi model

Brian A. Camley; Wei Wang

arxiv: 2604.15481 · v1 · submitted 2026-04-16 · ❄️ cond-mat.soft · physics.bio-ph· q-bio.QM

Divergence of detachment forces in the finite Voronoi model

Wei Wang , Brian A. Camley This is my paper

Pith reviewed 2026-05-10 09:24 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.bio-phq-bio.QM

keywords finite Voronoi modeldetachment forcestissue fracturecell adhesionnonconfluent monolayersline tensionregularizationsimulation artifact

0 comments

The pith

The finite Voronoi model develops diverging detachment forces that make tissue fracture times depend on the simulation time step.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper analyzes the finite Voronoi model for nonconfluent tissues, where cell boundaries use straight Voronoi edges plus circular arcs of fixed radius. When line tension at cell-medium interfaces exceeds cell-cell tension, detachment forces grow without bound, so smaller time steps can prevent clusters from rupturing in simulations. This artifact traces directly to the boundary geometry without a cutoff near separation. The authors add a simple regularization, calibrate the near-detachment behavior to a deformable polygon model, and show how parameters set the fracture timescale under two calibration choices. The issue matters for any simulation study of detachment, adhesion, or fracture in tissues that are not fully confluent.

Core claim

When the line tension on cell-medium interfaces exceeds the tension on cell-cell contacts, the finite Voronoi model exhibits a divergence of detachment forces. This produces strong time-step dependence in the fracture timescale of initially intact active clusters, with decreasing time step unphysically suppressing rupture events. The divergence arises from the model's use of straight Voronoi edges and fixed-radius circular arcs for boundaries. A simple regularization removes the artifact, after which the model is calibrated to a deformable polygon representation to study how physical parameters control fracture under two strategies.

What carries the argument

The geometric construction of cell boundaries from straight Voronoi edges and fixed-radius circular arcs, which produces diverging detachment forces as separation approaches the arc radius.

Load-bearing premise

The fixed-radius circular arcs and straight Voronoi edges remain an adequate geometric representation of cell boundaries all the way to the instant of detachment without additional physical regularization or cutoff.

What would settle it

Run simulations of active cell clusters with cell-medium tension larger than cell-cell tension and check whether fracture time increases without bound as the time step is made smaller; if rupture is increasingly suppressed, the divergence is confirmed.

Figures

Figures reproduced from arXiv: 2604.15481 by Brian A. Camley, Wei Wang.

**Figure 3.** Figure 3: Cell doublet. a, Schematic of a cell doublet with centers positioned symmetrically at r± = ±(ℓ − 𝜖)𝑥ˆ. The arc radius of each cell is ℓ, and the distance between the two cell centers is given by 𝑑 = 2(ℓ − 𝜖), thus the contact length (red) is 2√︁ ℓ 2 − (ℓ − 𝜖) 2 . b, Force between a cell doublet as a function of their separation distance for Λ = 0, 0.1, 0.2, 0.3, 0.4, and 0.5 (blue to red). Positive values … view at source ↗

**Figure 2.** Figure 2: Tissue cohesion is strongly dependent on simulation [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: Regularization by introducing a cutoff. a [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: Steady states of deformable polygon model. a [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Detachment of cell doublets in the finite Voronoi model (FV) and deformable polygon (DP) models. a [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 7.** Figure 7: Calibrating the finite Voronoi model (FV) to a deformable polygon (DP) model. a [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗

read the original abstract

Detachment and fracture are central to many tissue-level processes, but they are challenging to simulate with Voronoi-type models that typically assume a confluent tissue. Here we analyze the finite Voronoi model, a nonconfluent extension of conventional Voronoi models, in which cell boundaries are composed of straight Voronoi edges and circular arcs of fixed radius $\ell$. When the line tension on cell-medium interfaces exceeds the tension on cell-cell contacts, we find that the model exhibits a strong time-step dependence in the fracture timescale of initially intact active clusters: decreasing $\Delta t$ can unphysically suppress cluster rupture events. We trace this behavior to a divergence of detachment forces in the finite Voronoi model and introduce a simple regularization. Finally, we calibrate the near-detachment mechanics against a deformable polygon model and examine how key physical parameters control the tissue fracture timescale under two different calibration strategies. Our results show that, for studies focused on fracture or intercellular adhesion in nonconfluent monolayers, a physically motivated calibration of near-detachment mechanics in the finite Voronoi model is essential.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper identifies a real geometric divergence in detachment forces in the finite Voronoi model when cell-medium tension exceeds cell-cell tension, which creates unphysical timestep dependence in fracture times, and it supplies a regularization plus calibration fix.

read the letter

The central result is that the finite Voronoi construction (Voronoi edges plus fixed-radius circular arcs) produces diverging forces on cell-medium interfaces as cells approach detachment whenever the medium tension is larger than cell-cell tension. This makes the time to rupture of an active cluster shrink as the timestep gets smaller, which is an artifact. The authors trace it directly to the arc geometry and show how a simple regularization removes the divergence. They then calibrate the near-detachment regime against a deformable-polygon reference and test how fracture timescale depends on arc radius and tension ratio under two different calibration choices. That is the new technical contribution: a documented limitation plus a practical workaround for people running nonconfluent tissue simulations. It is useful because many groups rely on these models for fracture and adhesion studies in developmental biology and tissue engineering. The evidence for the divergence itself comes from the model equations rather than data fitting, and the calibration step uses an external model as benchmark, so circularity stays low. The main soft spot is that the regularization is ad hoc and the calibration is still internal to simulation; nothing here is checked against experiment, so users will still need to decide whether the regularized behavior matches their physical system. The fixed-arc assumption is acknowledged as the point that breaks without the fix. This paper is for researchers who already work with finite Voronoi or similar vertex models and need reliable detachment mechanics. It does not change the broader field but removes a hidden numerical pitfall. The thinking is clear and the argument is self-contained. I would send it for peer review.

Referee Report

0 major / 3 minor

Summary. The paper analyzes the finite Voronoi model for nonconfluent cell monolayers, in which boundaries consist of straight Voronoi edges and fixed-radius circular arcs of length ℓ. It reports that when cell-medium line tension exceeds cell-cell tension, detachment forces diverge, producing strong unphysical dependence of fracture timescale on integration time step Δt that can suppress rupture events. The authors trace the divergence to the model geometry, introduce a regularization, calibrate near-detachment mechanics against a deformable-polygon reference model, and examine how key parameters control fracture timescale under two calibration strategies.

Significance. If the central results hold, the work is significant for the cell-mechanics simulation community because it identifies and mitigates a numerical artifact that directly affects the reliability of finite Voronoi models when used for detachment and fracture studies. Explicit demonstration of the Δt-dependent suppression, together with the regularization and the calibration to an independent polygon model, supplies concrete guidance for model users. The authors are credited for keeping the argument self-contained, for grounding the fix in an external benchmark, and for exploring parameter dependence under multiple calibration choices.

minor comments (3)

The abstract refers to 'two different calibration strategies' without naming them; the main text should introduce and label these strategies (e.g., in §3 or §4) so that readers can follow the subsequent parameter sweeps without backtracking.
Notation for the fixed arc radius ℓ and the two line tensions should be defined once in the introduction and then used consistently; a short table of symbols with units would improve clarity.
Figure captions for the time-step dependence plots should explicitly list the Δt values employed and state whether the regularization is active, to allow immediate visual assessment of the artifact.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work on the finite Voronoi model and for recommending minor revision. The review accurately summarizes the key findings regarding detachment force divergence and the proposed regularization and calibration approaches.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from model geometry

full rationale

The paper derives the detachment force divergence mathematically from the finite Voronoi construction (straight edges + fixed-radius arcs) when cell-medium tension exceeds cell-cell tension; this is an explicit geometric consequence shown in the equations, not a fit or redefinition. Regularization is introduced as a fix, and near-detachment calibration uses an independent deformable-polygon reference model as benchmark. No self-citations are load-bearing for the central claim, no ansatz is smuggled, and no prediction reduces to its own input by construction. The argument about time-step dependence and calibration necessity stands on its own internal consistency.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on standard Voronoi geometry plus the ad-hoc choice of fixed arc radius ℓ and the assumption that line tensions are constant up to the detachment point.

free parameters (2)

arc radius ℓ
Fixed length scale for circular arcs at cell-medium boundaries; chosen to represent cell curvature but not derived from first principles.
line tensions (cell-cell vs cell-medium)
Two independent tension values whose relative magnitude triggers the divergence; fitted or chosen to match biological conditions.

axioms (2)

domain assumption Cell boundaries can be represented by straight Voronoi edges joined to circular arcs of constant radius without loss of essential mechanics near detachment.
Invoked throughout the model definition and force calculation.
domain assumption Force balance on cell vertices remains valid even as the contact angle approaches the singular configuration.
Underlying the derivation of the divergent force.

pith-pipeline@v0.9.0 · 5485 in / 1444 out tokens · 45683 ms · 2026-05-10T09:24:16.448658+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

[1]

Details of finite Voronoi model force calculations In the standard/finite Voronoi model, as given in Refs. [5, 11], an inner vertex connecting three cells𝑖,𝑗, and𝑘is given by hin =𝛼 𝑖r𝑖 +𝛼 𝑗r 𝑗 +𝛼 𝑘r𝑘 .(A1) The three barycentric coordinates of the circumcenter are 𝛼𝑖 =|r 𝑗 −r 𝑘 |2 (r𝑖 −r 𝑗 ) · (r𝑖 −r 𝑘)/𝐷,(A2a) 𝛼 𝑗 =|r 𝑖 −r 𝑘 |2 (r 𝑗 −r 𝑖) · (r 𝑗 −r 𝑘)/𝐷,...

work page
[2]

Details of deformable polygon model In our simulations of the deformable polygon model, each cell is represented by𝑀=100 vertices. Enumerating the vertices{h 𝑚}of cell𝑖in clockwise order, the force exerted on vertex𝑚is given by f𝑚 =− 𝜕𝐸 𝜕h𝑚 .(A13) We need the derivative of area𝐴 𝑖 with respect toh 𝑚 [11]: 𝜕 𝐴𝑖 𝜕h𝑚 = (h𝑚−1 −h 𝑚+1) ×ˆ𝑧 2 ,(A14) and derivati...

work page
[3]

K. K. Youssef and M. A. Nieto, Epithelial–mesenchymal transi- tion in tissue repair and degeneration, Nature Reviews Molecular Cell Biology25, 720 (2024)

work page 2024
[4]

Tripathi, H

S. Tripathi, H. Levine, and M. K. Jolly, The physics of cellu- lar decision making during epithelial–mesenchymal transition, Annual Review of Biophysics49, 1 (2020)

work page 2020
[5]

K. L. Harper, M. S. Sosa, D. Entenberg, H. Hosseini, J. F. Cheung, R. Nobre, A. Avivar-Valderas, C. Nagi, N. Girnius, R. J. Davis,et al., Mechanism of early dissemination and metastasis in her2+ mammary cancer, Nature540, 588 (2016)

work page 2016
[6]

K. J. Cheung and A. J. Ewald, A collective route to metastasis: Seeding by tumor cell clusters, Science352, 167 (2016)

work page 2016
[7]

D. Bi, X. Yang, M. C. Marchetti, and M. L. Manning, Motility- driven glass and jamming transitions in biological tissues, Phys. Rev. X6, 021011 (2016)

work page 2016
[8]

D. Bi, J. Lopez, J. M. Schwarz, and M. L. Manning, A density- independent rigidity transition in biological tissues, Nature Physics11, 1074 (2015)

work page 2015
[9]

Y. Chen, Q. Gao, J. Li, F. Mao, R. Tang, and H. Jiang, Activation of topological defects induces a brittle-to-ductile transition in epithelial monolayers, Phys. Rev. Lett.128, 018101 (2022)

work page 2022
[10]

Graner and Y

F. Graner and Y. Sawada, Can surface adhesion drive cell rear- rangement? Part II: a geometrical model, Journal of theoretical biology164, 477 (1993)

work page 1993
[11]

M. Bock, A. K. Tyagi, J.-U. Kreft, and W. Alt, Generalized Voronoi tessellation as a model of two-dimensional cell tissue dynamics, Bulletin of mathematical biology72, 1696 (2010). 11

work page 2010
[12]

Schaller and M

G. Schaller and M. Meyer-Hermann, Multicellular tumor spheroid in an off-lattice Voronoi-Delaunay cell model, Phys. Rev. E71, 051910 (2005)

work page 2005
[13]

Teomy, D

E. Teomy, D. A. Kessler, and H. Levine, Confluent and noncon- fluent phases in a model of cell tissue, Phys. Rev. E98, 042418 (2018)

work page 2018
[14]

Huang, H

J. Huang, H. Levine, and D. Bi, Bridging the gap between col- lective motility and epithelial-mesenchymal transitions through the active finite Voronoi model, Soft Matter19, 9389 (2023)

work page 2023
[15]

Nonomura, Study on multicellular systems using a phase field model, PloS one7, e33501 (2012)

M. Nonomura, Study on multicellular systems using a phase field model, PloS one7, e33501 (2012)

work page 2012
[16]

Palmieri, Y

B. Palmieri, Y. Bresler, D. Wirtz, and M. Grant, Multiple scale model for cell migration in monolayers: Elastic mismatch be- tween cells enhances motility, Scientific reports5, 11745 (2015)

work page 2015
[17]

Chiang, A

M. Chiang, A. Hopkins, B. Loewe, D. Marenduzzo, and M. C. Marchetti, Multiphase field model of cells on a substrate: From three dimensional to two dimensional, Phys. Rev. E110, 044403 (2024)

work page 2024
[18]

W. Wang, R. A. Law, E. Perez Ipi ˜na, K. Konstantopoulos, and B. A. Camley, Confinement, jamming, and adhesion in cancer cells dissociating from a collectively invading strand, PRX Life 3, 013012 (2025)

work page 2025
[19]

S. A. Sandersius and T. J. Newman, Modeling cell rheology with the subcellular element model, Physical biology5, 015002 (2008)

work page 2008
[20]

C. R. Sweet, S. Chatterjee, Z. Xu, K. Bisordi, E. D. Rosen, and M. Alber, Modelling platelet–blood flow interaction using the subcellular element langevin method, Journal of The Royal Society Interface8, 1760 (2011)

work page 2011
[21]

Sandersius, C

S. Sandersius, C. J. Weijer, and T. J. Newman, Emergent cell and tissue dynamics from subcellular modeling of active biome- chanical processes, Physical biology8, 045007 (2011)

work page 2011
[22]

Boromand, A

A. Boromand, A. Signoriello, F. Ye, C. S. O’Hern, and M. D. Shattuck, Jamming of deformable polygons, Phys. Rev. Lett. 121, 248003 (2018)

work page 2018
[23]

Lv, P.-C

J.-Q. Lv, P.-C. Chen, Y.-P. Chen, H.-Y. Liu, S.-D. Wang, J. Bai, C.-L. Lv, Y. Li, Y. Shao, X.-Q. Feng, and B. Li, Active hole formation in epithelioid tissues, Nature Physics20, 1313 (2024)

work page 2024
[24]

S. Weng, R. J. Huebner, and J. B. Wallingford, Convergent exten- sion requires adhesion-dependent biomechanical integration of cell crawling and junction contraction, Cell Reports39, 110666 (2022)

work page 2022
[25]

D. M. Sussman and M. Merkel, No unjamming transition in a Voronoi model of biological tissue, Soft matter14, 3397 (2018)

work page 2018
[26]

Lawson-Keister, T

E. Lawson-Keister, T. Zhang, F. Nazari, F. Fagotto, and M. L. Manning, Differences in boundary behavior in the 3d vertex and Voronoi models, PLoS computational biology20, e1011724 (2024)

work page 2024
[27]

H. Yue, C. R. Packard, and D. M. Sussman, Scale-dependent sharpening of interfacial fluctuations in shape-based models of dense cellular sheets, Soft Matter20, 9444 (2024)

work page 2024
[28]

Wang and B

W. Wang and B. A. Camley, wwang721/pyafv, Zen- odo.18091659 (2026), [version to be pinned]

work page 2026
[29]

Wang and B

W. Wang and B. A. Camley, Controlling tissue size by active fracture, Phys. Rev. E113, 034405 (2026)

work page 2026
[30]

E. L. Kaplan and P. Meier, Nonparametric estimation from in- complete observations, Journal of the American Statistical As- sociation53, 457 (1958)

work page 1958
[31]

A. G. Fletcher, M. Osterfield, R. E. Baker, and S. Y. Shvartsman, Vertex models of epithelial morphogenesis, Biophysical journal 106, 2291 (2014)

work page 2014
[32]

Spahn and R

P. Spahn and R. Reuter, A vertex model of Drosophila ventral furrow formation, PloS one8, e75051 (2013)

work page 2013
[33]

Smeets, M

B. Smeets, M. Cuvelier, J. Pe ˇsek, and H. Ramon, The effect of cortical elasticity and active tension on cell adhesion mechanics, Biophysical journal116, 930 (2019)

work page 2019
[34]

Vangheel, H

J. Vangheel, H. Ramon, and B. Smeets, Rigidity transitions in a three-dimensional active foam model of cell monolayers with frictional contact interactions, Phys. Rev. Res.8, 013022 (2026)

work page 2026
[35]

Y.-S. Chu, S. Dufour, J. P. Thiery, E. Perez, and F. Pincet, Johnson-Kendall-Roberts theory applied to living cells, Phys. Rev. Lett.94, 028102 (2005)

work page 2005
[36]

S. W. Byers, C. L. Sommers, B. Hoxter, A. M. Mercurio, and A. Tozeren, Role of E-cadherin in the response of tumor cell aggregates to lymphatic, venous and arterial flow: measurement of cell-cell adhesion strength, Journal of cell science108, 2053 (1995)

work page 2053
[37]

Rozema, C

D. Rozema, C. Fagotto-Kaufmann, A. Ruppel, P. Lasko, and F. Fagotto, Remodeling of cadherin contacts in embryonic mes- enchymal tissues during differential cell migration, Develop- mental Cell60, 2915 (2025)

work page 2025
[38]

Eckert, V

J. Eckert, V. Matylitskaya, S. Kasemann, S. Partel, and T. Schmidt, Cell–cell separation device: A new approach to measuring intercellular detachment forces, Review of Scientific Instruments96, 095002 (2025)

work page 2025
[39]

P. E. Kloeden and E. Platen,Numerical Solution of Stochastic Differential Equations(Springer, 1992)

work page 1992

[1] [1]

Details of finite Voronoi model force calculations In the standard/finite Voronoi model, as given in Refs. [5, 11], an inner vertex connecting three cells𝑖,𝑗, and𝑘is given by hin =𝛼 𝑖r𝑖 +𝛼 𝑗r 𝑗 +𝛼 𝑘r𝑘 .(A1) The three barycentric coordinates of the circumcenter are 𝛼𝑖 =|r 𝑗 −r 𝑘 |2 (r𝑖 −r 𝑗 ) · (r𝑖 −r 𝑘)/𝐷,(A2a) 𝛼 𝑗 =|r 𝑖 −r 𝑘 |2 (r 𝑗 −r 𝑖) · (r 𝑗 −r 𝑘)/𝐷,...

work page

[2] [2]

Details of deformable polygon model In our simulations of the deformable polygon model, each cell is represented by𝑀=100 vertices. Enumerating the vertices{h 𝑚}of cell𝑖in clockwise order, the force exerted on vertex𝑚is given by f𝑚 =− 𝜕𝐸 𝜕h𝑚 .(A13) We need the derivative of area𝐴 𝑖 with respect toh 𝑚 [11]: 𝜕 𝐴𝑖 𝜕h𝑚 = (h𝑚−1 −h 𝑚+1) ×ˆ𝑧 2 ,(A14) and derivati...

work page

[3] [3]

K. K. Youssef and M. A. Nieto, Epithelial–mesenchymal transi- tion in tissue repair and degeneration, Nature Reviews Molecular Cell Biology25, 720 (2024)

work page 2024

[4] [4]

Tripathi, H

S. Tripathi, H. Levine, and M. K. Jolly, The physics of cellu- lar decision making during epithelial–mesenchymal transition, Annual Review of Biophysics49, 1 (2020)

work page 2020

[5] [5]

K. L. Harper, M. S. Sosa, D. Entenberg, H. Hosseini, J. F. Cheung, R. Nobre, A. Avivar-Valderas, C. Nagi, N. Girnius, R. J. Davis,et al., Mechanism of early dissemination and metastasis in her2+ mammary cancer, Nature540, 588 (2016)

work page 2016

[6] [6]

K. J. Cheung and A. J. Ewald, A collective route to metastasis: Seeding by tumor cell clusters, Science352, 167 (2016)

work page 2016

[7] [7]

D. Bi, X. Yang, M. C. Marchetti, and M. L. Manning, Motility- driven glass and jamming transitions in biological tissues, Phys. Rev. X6, 021011 (2016)

work page 2016

[8] [8]

D. Bi, J. Lopez, J. M. Schwarz, and M. L. Manning, A density- independent rigidity transition in biological tissues, Nature Physics11, 1074 (2015)

work page 2015

[9] [9]

Y. Chen, Q. Gao, J. Li, F. Mao, R. Tang, and H. Jiang, Activation of topological defects induces a brittle-to-ductile transition in epithelial monolayers, Phys. Rev. Lett.128, 018101 (2022)

work page 2022

[10] [10]

Graner and Y

F. Graner and Y. Sawada, Can surface adhesion drive cell rear- rangement? Part II: a geometrical model, Journal of theoretical biology164, 477 (1993)

work page 1993

[11] [11]

M. Bock, A. K. Tyagi, J.-U. Kreft, and W. Alt, Generalized Voronoi tessellation as a model of two-dimensional cell tissue dynamics, Bulletin of mathematical biology72, 1696 (2010). 11

work page 2010

[12] [12]

Schaller and M

G. Schaller and M. Meyer-Hermann, Multicellular tumor spheroid in an off-lattice Voronoi-Delaunay cell model, Phys. Rev. E71, 051910 (2005)

work page 2005

[13] [13]

Teomy, D

E. Teomy, D. A. Kessler, and H. Levine, Confluent and noncon- fluent phases in a model of cell tissue, Phys. Rev. E98, 042418 (2018)

work page 2018

[14] [14]

Huang, H

J. Huang, H. Levine, and D. Bi, Bridging the gap between col- lective motility and epithelial-mesenchymal transitions through the active finite Voronoi model, Soft Matter19, 9389 (2023)

work page 2023

[15] [15]

Nonomura, Study on multicellular systems using a phase field model, PloS one7, e33501 (2012)

M. Nonomura, Study on multicellular systems using a phase field model, PloS one7, e33501 (2012)

work page 2012

[16] [16]

Palmieri, Y

B. Palmieri, Y. Bresler, D. Wirtz, and M. Grant, Multiple scale model for cell migration in monolayers: Elastic mismatch be- tween cells enhances motility, Scientific reports5, 11745 (2015)

work page 2015

[17] [17]

Chiang, A

M. Chiang, A. Hopkins, B. Loewe, D. Marenduzzo, and M. C. Marchetti, Multiphase field model of cells on a substrate: From three dimensional to two dimensional, Phys. Rev. E110, 044403 (2024)

work page 2024

[18] [18]

W. Wang, R. A. Law, E. Perez Ipi ˜na, K. Konstantopoulos, and B. A. Camley, Confinement, jamming, and adhesion in cancer cells dissociating from a collectively invading strand, PRX Life 3, 013012 (2025)

work page 2025

[19] [19]

S. A. Sandersius and T. J. Newman, Modeling cell rheology with the subcellular element model, Physical biology5, 015002 (2008)

work page 2008

[20] [20]

C. R. Sweet, S. Chatterjee, Z. Xu, K. Bisordi, E. D. Rosen, and M. Alber, Modelling platelet–blood flow interaction using the subcellular element langevin method, Journal of The Royal Society Interface8, 1760 (2011)

work page 2011

[21] [21]

Sandersius, C

S. Sandersius, C. J. Weijer, and T. J. Newman, Emergent cell and tissue dynamics from subcellular modeling of active biome- chanical processes, Physical biology8, 045007 (2011)

work page 2011

[22] [22]

Boromand, A

A. Boromand, A. Signoriello, F. Ye, C. S. O’Hern, and M. D. Shattuck, Jamming of deformable polygons, Phys. Rev. Lett. 121, 248003 (2018)

work page 2018

[23] [23]

Lv, P.-C

J.-Q. Lv, P.-C. Chen, Y.-P. Chen, H.-Y. Liu, S.-D. Wang, J. Bai, C.-L. Lv, Y. Li, Y. Shao, X.-Q. Feng, and B. Li, Active hole formation in epithelioid tissues, Nature Physics20, 1313 (2024)

work page 2024

[24] [24]

S. Weng, R. J. Huebner, and J. B. Wallingford, Convergent exten- sion requires adhesion-dependent biomechanical integration of cell crawling and junction contraction, Cell Reports39, 110666 (2022)

work page 2022

[25] [25]

D. M. Sussman and M. Merkel, No unjamming transition in a Voronoi model of biological tissue, Soft matter14, 3397 (2018)

work page 2018

[26] [26]

Lawson-Keister, T

E. Lawson-Keister, T. Zhang, F. Nazari, F. Fagotto, and M. L. Manning, Differences in boundary behavior in the 3d vertex and Voronoi models, PLoS computational biology20, e1011724 (2024)

work page 2024

[27] [27]

H. Yue, C. R. Packard, and D. M. Sussman, Scale-dependent sharpening of interfacial fluctuations in shape-based models of dense cellular sheets, Soft Matter20, 9444 (2024)

work page 2024

[28] [28]

Wang and B

W. Wang and B. A. Camley, wwang721/pyafv, Zen- odo.18091659 (2026), [version to be pinned]

work page 2026

[29] [29]

Wang and B

W. Wang and B. A. Camley, Controlling tissue size by active fracture, Phys. Rev. E113, 034405 (2026)

work page 2026

[30] [30]

E. L. Kaplan and P. Meier, Nonparametric estimation from in- complete observations, Journal of the American Statistical As- sociation53, 457 (1958)

work page 1958

[31] [31]

A. G. Fletcher, M. Osterfield, R. E. Baker, and S. Y. Shvartsman, Vertex models of epithelial morphogenesis, Biophysical journal 106, 2291 (2014)

work page 2014

[32] [32]

Spahn and R

P. Spahn and R. Reuter, A vertex model of Drosophila ventral furrow formation, PloS one8, e75051 (2013)

work page 2013

[33] [33]

Smeets, M

B. Smeets, M. Cuvelier, J. Pe ˇsek, and H. Ramon, The effect of cortical elasticity and active tension on cell adhesion mechanics, Biophysical journal116, 930 (2019)

work page 2019

[34] [34]

Vangheel, H

J. Vangheel, H. Ramon, and B. Smeets, Rigidity transitions in a three-dimensional active foam model of cell monolayers with frictional contact interactions, Phys. Rev. Res.8, 013022 (2026)

work page 2026

[35] [35]

Y.-S. Chu, S. Dufour, J. P. Thiery, E. Perez, and F. Pincet, Johnson-Kendall-Roberts theory applied to living cells, Phys. Rev. Lett.94, 028102 (2005)

work page 2005

[36] [36]

S. W. Byers, C. L. Sommers, B. Hoxter, A. M. Mercurio, and A. Tozeren, Role of E-cadherin in the response of tumor cell aggregates to lymphatic, venous and arterial flow: measurement of cell-cell adhesion strength, Journal of cell science108, 2053 (1995)

work page 2053

[37] [37]

Rozema, C

D. Rozema, C. Fagotto-Kaufmann, A. Ruppel, P. Lasko, and F. Fagotto, Remodeling of cadherin contacts in embryonic mes- enchymal tissues during differential cell migration, Develop- mental Cell60, 2915 (2025)

work page 2025

[38] [38]

Eckert, V

J. Eckert, V. Matylitskaya, S. Kasemann, S. Partel, and T. Schmidt, Cell–cell separation device: A new approach to measuring intercellular detachment forces, Review of Scientific Instruments96, 095002 (2025)

work page 2025

[39] [39]

P. E. Kloeden and E. Platen,Numerical Solution of Stochastic Differential Equations(Springer, 1992)

work page 1992