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REVIEW 2 major objections 6 minor 61 references

Traction force microscopy for linear and nonlinear elastic materials as a parameter identification inverse problem

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper shows that traction force microscopy can be formulated as a parameter identification inverse problem whose nonlinear forward operator is Fréchet-differentiable with a self-adjoint derivative, enabling stable regularized…

desk verdict Solid nonlinear TFM framework with a real regularity gap in the well-posedness claim; worth refereeing, not desk-rejecting. read the letter →

arxiv 2411.19917 v1 pith:5QUU4DOB submitted 2024-11-29 math.NA cs.NA

Gesa Sarnighausen , Tram Thi Ngoc Nguyen , Thorsten Hohage , Mangalika Sinha , Sarah Koester , Timo Betz , Ulrich Sebastian Schwarz , Anne Wald This is my paper · ORCID
classification math.NAcs.NA MSC 65J2035Q7474B2074G7565N21
keywords tractionforcemicroscopyinverseproblemsparameteridentificationnonlinearelasticityhyperelasticitypolyconvexstoredenergyFréchetderivativeregularization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to put traction force microscopy (TFM) on a rigorous inverse-problem footing. It treats linear 2.5D TFM and nonlinear pure 2D TFM as parameter identification problems, where the unknown cell force is recovered from measured substrate displacement. The central claim is that the nonlinear force-to-displacement map is well defined on suitable Banach spaces, Fréchet-differentiable, and invertible by regularized Newton-CG, provided the substrate follows a specifically chosen polyconvex energy law. A working numerical method follows, including reconstructions from experimental data, which matters because nonlinear substrates under large deformation cannot be handled by the analytical Boussinesq-based methods used in standard TFM.

What carries the argument

The load-bearing object is the forward operator $S$ together with the stored energy function (20), a two-dimensional polyconvex Ogden-type hyperelastic law $W(F) = \frac{\mu}{2}|F|^2 + \frac{\lambda}{4}(\det F)^2 - (\mu + \frac{\lambda}{2})\ln(\det F) - \frac{3\mu}{2} - \frac{\lambda}{4}$. Polyconvexity and coercivity allow the displacement boundary value problem to be solved by minimizing the energy $G(u) = \int_\Omega W(I + \nabla u)\,dx - \int_\Omega T u\,dx$, and the same law is engineered so that its linearization matches Hooke's law near a natural state. The Fréchet derivative $S'(T)h = v$ solves the linearized elasticity system in (27), and its $L^2$-self-adjointness means the adjoint can be evaluated by solving the same type of boundary value problem, which makes the adjoint-based Newton-CG iteration practical.

What would settle it

Compute, on a domain with a nonsmooth corner, all energy-minimizing solutions of the displacement problem with $W$ from (20) for a fixed $T \in L^p$; finding two distinct minimizers, or a minimizer whose second derivatives fail to be integrable, would break the single-valuedness and differentiability of the nonlinear forward operator.

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Extended reading notes

Core claim

For the nonlinear pure 2D model, the paper claims that the parameter-to-state map $S: T \mapsto u$ defined by the boundary value problem $-\operatorname{div}(\sigma(u)) = T$ in $\Omega$, $u = 0$ on $\partial\Omega$, with stress derived from the stored energy function $W(F) = \frac{\mu}{2}|F|^2 + \frac{\lambda}{4}(\det F)^2 - (\mu + \frac{\lambda}{2})\ln(\det F) - \frac{3\mu}{2} - \frac{\lambda}{4}$, is a well-defined forward operator. The paper proves existence of energy minimizers (Theorem 3.3), local uniqueness of admissible solutions and bounded invertibility of the linearized operator (Theorem 3.4), polyconvexity and coercivity of $W$ (Theorem 3.5), and an explicit Fréchet derivative (Theorem 3.7) that is self-adjoint in $L^2$ (Theorem 3.8). These results justify applying discrepancy-principle-truncated Newton-CG with an adjoint-based inner iteration to recover force densities. For the linear 2.5D case, the forward operator is a bounded linear map whose adjoint is obtained through an auxiliary elasticity problem, and the paper demonstrates reconstructions from simulated and experimental displacement data; for a high-force phantom, using the linear reconstruction as an initial guess reduces the nonlinear reconstruction error from roughly 7% to below 3%. The nonlinear theory depends on a regularity bridge: the energy minimizer from Theorem 3.3 lies in $W^{1,p}$, while the uniqueness and derivative theorems require a $W^{2,p}$ admissible state, a step the paper states is still needed and points to a known regularity route.

Load-bearing premise

The nonlinear theory stands on the unproven bridge that the energy-minimizing displacement, which exists in the Sobolev space $W^{1,p}$, is actually regular enough (in $W^{2,p}$ and admissible) for the local-uniqueness and derivative theorems to apply; the paper says this regularity still has to be improved and cites a possible route.

Editorial extensions

If this is right

  • Nonlinear TFM with hyperelastic substrates under large deformation can be solved by iterative regularization rather than by numerical differentiation of noisy images.
  • The same framework supplies a linear pure 2D solver; applying it to measured fibroblast data yields traction maps in the same range as the standard Fourier method and suggests an effective substrate thickness near $1\,\mu$m.
  • Because the derivative is self-adjoint, computing the adjoint for the Newton-CG inner iteration costs one linearized elasticity solve, so the extra expense of nonlinear inversion over linear inversion is moderate.
  • Choosing $H^1_0$ as the parameter space smooths high-noise reconstructions, whereas $L^2$ gives lower errors on cleaner data, providing a practical norm-selection rule.
  • The analytical framework extends to other polyconvex stored energy functions, so material laws can be swapped without redoing the inverse-problem architecture.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One consequence the paper leaves implicit is that the same adjoint-based machinery would allow joint estimation of the effective thickness $h$ and the traction field $T$, since the conversion $T = t/h$ is linear in $t$ and $h$ enters only as a scalar; the reported experimental value $h \approx 1\,\mu$m indicates the data carry information about it.
  • If Morrey-type regularity could be established for the specific energy (20), the local character of Theorem 3.4 would become global on a full neighborhood of admissible data, and the conditional stability constants of the linearized operator would likely determine convergence rates for the Newton-CG iteration, which the paper does not compute.
  • The reported norm-selection behavior suggests a practical stopping-rule design: monitor the discrepancy-principle iterate under both $L^2$ and $H^1_0$ penalties and prefer the $H^1_0$ reconstruction when the noise estimate exceeds a few percent; the paper documents the effect but does not propose this rule.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. This paper reformulates traction force microscopy as a parameter identification problem. For linear 2.5D TFM, it defines the forward operator A from boundary traction stresses to interior displacements via the weak mixed boundary value problem (8)-(9), proves well-posedness, and derives the adjoint (Lemma 2.2). For nonlinear pure 2D TFM, it introduces the stored energy function (20) from the polyconvex Ogden family, proves coercivity and polyconvexity under the parameter condition lambda > 2mu/(e-1) (Theorem 3.5), states an existence result for minimizers in W^{1,p} (Theorem 3.3), and states a local uniqueness theorem requiring admissible W^{2,p} solutions (Theorem 3.4). It then asserts that the parameter-to-state map S in (13) is well-defined, locally unique, and Frechet-differentiable with a computable self-adjoint derivative (Theorems 3.7 and 3.8), and it reports numerical experiments on simulated and experimental data, including a comparison with the standard FTTC method.

Significance. If the theoretical claims were fully established, the paper would provide a useful functional-analytic framework for nonlinear TFM, with explicit derivative and adjoint formulas that allow the use of standard regularization algorithms, and an open implementation. The linear 2.5D analysis is sound, the constitutive derivation in Theorem 3.5 is careful and explicit, and the Frechet derivative and adjoint computations in Theorems 3.7 and 3.8 are detailed and implementable. The availability of code and data is a clear strength. However, the central well-posedness claim for the nonlinear forward operator is not supported because of the regularity gap described in the major comments; the stress-test concern lands. Since this gap is load-bearing for the main theoretical contribution, the paper in its current form cannot be accepted as a rigorous mathematical treatment of nonlinear TFM.

major comments (2)
  1. [Sections 3.2.2 and 3.3 (Remark 3.6)] The existence result Theorem 3.3 supplies minimizers only in W^{1,p}(Omega,R^2), whereas Theorem 3.4 and the subsequent analysis require a solution in W^{2,p}(Omega) intersect W^{1,p}_0(Omega). The manuscript itself states in Section 3.2.2 that 'the regularity of this solution has to be improved ... This might be done in a similar manner as described in [37]' and that 'no deformation state u in W^{1,p}(Omega,R^2) can be admissible.' Remark 3.6 nevertheless asserts, without proof, that the conditions of Theorem 3.5 for p=s>2 imply 'at least one solution u in W^{2,p}(Omega,R^2) intersect W^{1,p}_0(Omega,R^2).' Polyconvexity and coercivity alone do not yield this higher regularity, and the cited Morrey regularity result is not applied to the present minimization problem. Because S in (13) is defined through this solution and because Theorems 3.7 and 3.8 both rely on Theorem 3.4 via the Implicit Function Theorem, the well-definedness and differentiability of S are not established. This is a load-bearing gap in the paper's central theoretical claim.
  2. [Section 3.2.2, Theorem 3.4] Even if a W^{2,p} solution were known to exist for every T, Theorem 3.4 only provides local uniqueness in a neighborhood V(bar u) x W(bar T) of an admissible pair. Remark 3.6 then selects X=L^p(Omega,R^2) (or H^1_0(Omega,R^2)) without restricting the parameter domain to such a neighborhood. Without a global uniqueness statement, the map S in (13) may be multi-valued on the chosen space X, and the inverse problem as posed on the full space may be ill-posed. The paper should either prove global uniqueness, or formulate S and the derivative theorems as local statements on an explicitly defined open set of admissible data.
minor comments (6)
  1. [Theorem 3.4] The notation 'V(bar u) in W^{2,p}(Omega) intersect W^{1,p}_0(Omega)' and 'W(bar T) in L^p(Omega)' should read 'V(bar u) subset W^{2,p}(Omega) intersect W^{1,p}_0(Omega)' and 'W(bar T) subset L^p(Omega),' since these are neighborhoods, not elements of the spaces.
  2. [Section 3.2.2] The statement that 'no deformation state u in W^{1,p}(Omega,R^2) can be admissible' is surprising in view of the Sobolev embedding W^{1,p}(Omega,R^2) into C^0 for p>2 used in the same paragraph; the precise notion of admissibility and the required regularity threshold should be clarified.
  3. [Section 4.3, Tables 2 and 3] The conclusions about the L^2-penalty being better for low noise and the H^1_0-penalty being better for high noise are based on only two force fields and two noise levels; in Table 3 the L^2 reconstruction at 15.63% noise has an error of 82.93%, which is close to a trivial reconstruction, so the comparative claim should be tempered.
  4. [Theorem 3.8] The statement that S'(T) is self-adjoint in L^2 should specify the domain of the unbounded operator (for example, the natural domain inherited from L^p(Omega,R^2) or W^{2,p}(Omega,R^2) intersect W^{1,p}_0(Omega,R^2)) and should note that the integration-by-parts argument is justified under the regularity assumptions; otherwise the term 'self-adjoint' is ambiguous.
  5. [Appendix A] The heading 'Proof of Lemma 3.3' should refer to Theorem 3.3, since the statement being proved is labeled as a theorem in the main text.
  6. [Throughout] There are minor typographical issues, such as the missing space in 'Lame constants lambda' before equation (5), the awkward piecewise formatting of equation (31), and the use of Euler's number e in condition (23) without an explicit definition; these are easy to correct.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the nonlinear forward operator analysis is self-contained against standard elasticity benchmarks, and the acknowledged W^{2,p} regularity gap is a correctness risk rather than a circular reduction.

full rationale

The central derivation chain is not circular. The linear 2.5D forward operator and its adjoint (Lemma 2.2) are proven inside the paper from Lax-Milgram and Korn's inequality. The nonlinear analysis rests on in-paper proofs of polyconvexity and coercivity for the stored energy (Theorem 3.5), an existence proof via the direct method adapted from Ball and Ciarlet (Theorem 3.3 and Appendix A), and a local uniqueness theorem (Theorem 3.4) that imports the Implicit Function Theorem argument from Ciarlet, Zeidler, and Valent. The Frechet derivative and self-adjointness results (Theorems 3.7 and 3.8) are then computed explicitly from the linearized PDE, not assumed. Self-citations, such as Blumberg-Schwarz for noise modeling and Schwarz-Soine for material parameters, are contextual and not load-bearing. The one genuine internal weakness is the regularity bridge in Section 3.2.2 and Remark 3.6: Theorem 3.3 yields only W^{1,p} minimizers, while Theorem 3.4 requires an admissible W^{2,p} starting point, and the paper explicitly says this regularity 'has to be improved' and 'might be done in a similar manner as described in [37]'. Remark 3.6 then asserts the W^{2,p} conclusion without proof. This is an unproven step and a correctness risk, but it is not circular: the paper does not define the conclusion into the assumptions, and the gap is openly flagged rather than hidden. The experimental statement that the reconstructed traction has the same data range as the FTTC solution and therefore indicates h approximately 1 micron is a qualitative comparison; no documented fitting of h to the FTTC output is described, so the fitted-input-as-prediction pattern is not established. Overall, the derivation is self-contained against external mathematical benchmarks, and the score is 0.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard mathematical tools (Korn, Lax-Milgram, Sobolev embeddings, implicit function theorem), standard TFM modeling assumptions (thin substrate, zero Dirichlet, thickness-averaged equilibrium), and one ad hoc material law. The decisive unproven premise is the W^{2,p} regularity bridge for the nonlinear minimizer. No new physical entities are introduced.

free parameters (3)
  • effective thickness h = h = 1 for simulations; h about 1 micrometer for the experimental dataset, inferred by matching the FTTC reconstruction
    The 2D model converts force density T to traction stress through T = t/h (Eq. 14). The value of h is required to interpret displacement data as traction; in simulations it is an arbitrary scale, and in Section 4.4 it is inferred from agreement with the standard FTTC method.
  • discrepancy principle parameter tau = tau = 1.2 and 1.01 (linear 2.5D), tau = 1.01 (nonlinear 2D), tau = 1.1 (experimental)
    Stopping criterion parameter chosen by hand (Tables 1 to 3, Section 4.4); standard practice, but affects reconstruction quality.
  • inner iteration damping factor rho = rho = 0.7
    Truncated Newton-CG inner stopping criterion (Section 4.3), chosen by hand.
assumptions (8)
  • standard math Korn's inequality and the Lax-Milgram lemma are used to prove well-posedness of the linear parameter-to-state map A (Section 2.2, Eqs. (9)-(10)).
    Standard functional analysis tools for linear elasticity boundary value problems.
  • standard math Sobolev embedding W^{1,p} into C^0 for p > 2, Rellich-Kondrachov compactness, Mazur's theorem, Fatou's lemma, and Riesz-Fischer are used in the existence proof (Appendix A).
    Standard tools for existence proofs via the direct method.
  • standard math Implicit Function Theorem and the Banach algebra property of W^{m,p} for mp > n are used for the Fréchet derivative and local uniqueness (Sections 3.2.2 and 3.4).
    Standard tools for local solvability and differentiability of parameter-dependent PDEs.
  • standard math Ball's existence theory for nonlinear elasticity (Theorem 6.2 of [5]) is invoked as an external foundational result (Section 3.2.1 and Appendix A).
    The paper adapts Ball's theorem to the 2D displacement BVP; the adaptation is sketched but relies on the original result.
  • domain assumption The pure 2D model assumes the substrate is larger than the support of the traction stresses, so zero Dirichlet boundary conditions apply (Section 3.1).
    Modeling assumption reflecting sparsely seeded cells; not valid for all experimental geometries.
  • domain assumption The thickness-averaged stress equilibrium condition (12), T = t/h, models the 3D substrate as a 2D medium with a constant effective thickness h (Section 3.1, from [15, 36]).
    Standard thin-substrate modeling assumption in TFM; the paper notes it differs from thick-substrate experiments.
  • ad hoc to paper The stored energy function (20) is chosen ad hoc within the polyconvex Ogden family to guarantee polyconvexity and coercivity; the condition lambda > 2 mu/(e-1) is imposed on material parameters (Theorem 3.5).
    This specific material law is introduced for this paper's theory, not a standard empirical law like neo-Hookean.
  • ad hoc to paper The W^{1,p} minimizer from Theorem 3.3 is assumed to have the W^{2,p} regularity and admissibility required by Theorem 3.4 (Section 3.2.2 and Remark 3.6).
    The paper explicitly leaves this regularity improvement to future work but uses it to define the forward operator.

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Pith. "Pith review of Traction force microscopy for linear and nonlinear elastic materials as a parameter identification inverse problem." pith.science (2026). https://pith.science/paper/5QUU4DOB

@misc{pith2026241119917,
  author       = {Pith},
  title        = {Pith review of: Traction force microscopy for linear and nonlinear elastic materials as a parameter identification inverse problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5QUU4DOB}},
  note         = {Machine review of arXiv:2411.19917}
}
read the original abstract

Traction force microscopy is a method widely used in biophysics and cell biology to determine forces that biological cells apply to their environment. In the experiment, the cells adhere to a soft elastic substrate, which is then deformed in response to cellular traction forces. The inverse problem consists in computing the traction stress applied by the cell from microscopy measurements of the substrate deformations. In this work, we consider a linear model, in which 3D forces are applied at a 2D interface, called 2.5D traction force microscopy, and a nonlinear pure 2D model, from which we directly obtain a linear pure 2D model. All models lead to a linear resp. nonlinear parameter identification problem for a boundary value problem of elasticity. We analyze the respective forward operators and conclude with some numerical experiments for simulated and experimental data.

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Reference graph

Works this paper leans on

61 extracted references · 59 canonical work pages

  1. [37]

    Michel, V

    R. Michel, V. Peschetola, G. Vitale, J. Etienne, A. Duperray, D. Ambrosi, L. Preziosi, and C. Verdier. Mathematical framework for Traction Force Microscopy. ESAIM Proceedings, 42, 12 2013

    work page 2013

  2. [1]

    Adams and J.J.F

    R.A. Adams and J.J.F. Fournier. Sobolev Spaces. ISSN. Elsevier Science, 2003

    2003

  3. [2]

    Ahmed, W.and Fodor and T

    ´E. Ahmed, W.and Fodor and T. Betz. Active cell mechanics: Measurement and theory. Biochimica et Biophysica Acta (BBA)-Molecular Cell Research , 1853(11):3083–3094, 2015

    work page 2015

  4. [3]

    D. Ambrosi. Cellular traction as an inverse problem. SIAM Journal on Applied Mathematics , 66(6):2049–2060, 2006

    work page 2006

  5. [4]

    Ambrosi, A

    D. Ambrosi, A. Duperray, V. Peschetola, and C. Verdier. Traction patterns of tumor cells. Journal of Mathematical Biology , 58:163–181, 2009

    work page 2009

  6. [5]

    J. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Archive for Rational Mechanics and Analysis , 63:337–403, 1976

    work page 1976

  7. [6]

    Banerjee, M

    S. Banerjee, M. Gardel, and U.S. Schwarz. The actin cytoskeleton as an active adaptive material. Annual review of condensed matter physics , 11(1):421–439, 2020

    work page 2020

  8. [7]

    Bar-Kochba, J

    E. Bar-Kochba, J. Toyjanova, E. Andrews, K.-S. Kim, and C. Franck. A Fast Iterative Digital Volume Correlation Algorithm for Large Deformations. Experimental Mechanics, 55, 08 2014

    work page 2014

Show all 61 references
  1. [8]

    Barrasa-Fano, A

    J. Barrasa-Fano, A. Shapeti, J. De Jong, A. Ranga, J. Sanz-Herrera, and H. Van Oosterwyck. Advanced in silico validation framework for three-dimensional traction force microscopy and application to an in vitro model of sprouting angiogenesis. Acta Biomaterialia, 126:326–338, 2021. 25

    2021

  2. [9]

    Blumberg and U.S

    J. Blumberg and U.S. Schwarz. Comparison of direct and inverse methods for 2.5D traction force microscopy. PLOS ONE , 17(1):1–25, 01 2022

    2022

  3. [10]

    Botti and L

    L. Botti and L. Verzeroli. BR2 discontinuous Galerkin methods for finite hyperelastic defor- mations. Journal of Computational Physics , 463:111303, 2022

    2022

  4. [11]

    A.F. Bower. Applied Mechanics of Solids . Taylor & Francis, 2009

    2009

  5. [12]

    B¨ uhler and D

    T. B¨ uhler and D. Salamon.Functional Analysis. Graduate studies in mathematics. American Mathematical Society, 2018

    2018

  6. [13]

    Butler, I

    J. Butler, I. Toli´ c, B. Fabry, and J. Fredberg. Traction fields, moments, and strain energy that cells exert on their surroundings. American journal of physiology. Cell physiology , 282:C595– 605, 04 2002

    2002

  7. [14]

    P.G. Ciarlet. Three-Dimensional Elasticity. Number Bd. 20 in Mathematical Elasticity. Else- vier Science, 1994

    1994

  8. [15]

    Dembo, T

    M. Dembo, T. Oliver, A. Ishihara, and K. Jacobson. Imaging the traction stresses exerted by locomoting cells with the elastic substratum method. Biophysical Journal , 70(4):2008–2022, 1996

    2008

  9. [16]

    Dembo and Y.-L

    M. Dembo and Y.-L. Wang. Stresses at the cell-to-substrate interface during locomotion of fibroblasts. Biophysical Journal, 76(4):2307–2316, 1999

    1999

  10. [17]

    Denisin, H

    A. Denisin, H. Kim, I. Riedel-Kruse, and B. Pruitt. Field Guide to Traction Force Microscopy. Cellular and Molecular Bioengineering , 17, 04 2024

    2024

  11. [18]

    Elstrodt

    J. Elstrodt. Maß- und Integrationstheorie. Springer, 8 edition, 2018

    2018

  12. [19]

    H. W. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems . Mathematics and Its Applications. Springer Dordrecht, 1996

    1996

  13. [20]

    Franck, S

    C. Franck, S. Hong, S. Maskarinec, D. Tirrell, and G. Ravichandran. Three-dimensional Full- field Measurements of Large Deformations in Soft Materials Using Confocal Microscopy and Digital Volume Correlation. Experimental Mechanics, 47:427–438, 05 2007

    2007

  14. [21]

    Hanke, D

    J. Hanke, D. Probst, A. Zemel, U.S. Schwarz, and S. K¨ oster. Dynamics of force generation by spreading platelets. Soft Matter , 14, 07 2018

    2018

  15. [22]

    M. Hanke. Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems. Numerical Functional Analysis and Optimization , 18:971–993, 1997

    1997

  16. [23]

    A. K. Harris, D. Stopak, and P. Wild. Fibroblast traction as a mechanism for collagen mor- phogenesis. Nature, 290(5803), March 1981

    1981

  17. [24]

    A. K. Harris, P. Wild, and D. Stopak. Silicone Rubber Substrata: A New Wrinkle in the Study of Cell Locomotion. Science, 208(4440):177–179, 1980

    1980

  18. [25]

    Hartmann and P

    S. Hartmann and P. Neff. Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. International Journal of Solids and Structures , 40(11):2767–2791, 2003

    2003

  19. [26]

    Hohage, P

    T. Hohage, P. Mickan, B. M¨ uller, F. Oberender, and C. R¨ ugge. regpy: Python tools for regularization methods, 2024

    2024

  20. [27]

    Holzapfel

    G.A. Holzapfel. Nonlinear Solid Mechanics: A Continuum Approach for Engineering . Wiley, 2000

    2000

  21. [28]

    S. Hur, J. del ´Alamo, J. Seok Park, Y.-S. Li, H. A. Nguyen, D. Teng, K.-C. Wang, L. Flores, 26 B. Alonso-Latorre, J. C. Lasheras, and S. Chien. Roles of cell confluency and fluid shear in 3-dimensional intracellular forces in endothelial cells. Proceedings of the National Aca...

    2012

  22. [29]

    S. Hur, Y. Zhao, Y.-S. Li, E. Botvinick, and S. Chien. Live Cells Exert 3-Dimensional Traction Forces on Their Substrata. Cellular and molecular bioengineering , 2:425–436, 09 2009

    2009

  23. [30]

    Metastatic cancer cells tenaciously indent impenetrable, soft substrates

    R Kristal-Muscal, Liron Dvir, and Daphne Weihs. Metastatic cancer cells tenaciously indent impenetrable, soft substrates. New Journal of Physics , 15(3):035022, 2013

    2013

  24. [31]

    Landau, L.P

    L.D. Landau, L.P. Pitaevskii, A.M. Kosevich, and E.M. Lifshitz. Theory of Elasticity: Volume

  25. [32]

    Elsevier Science, 1986

    Course of theoretical physics. Elsevier Science, 1986

    1986

  26. [33]

    Lekka, K

    M. Lekka, K. Gnanachandran, A. Kubiak, T. Zieli´ nski, and J. Zem la. Traction force microscopy - Measuring the forces exerted by cells. Micron, 150:103138, 2021

    2021

  27. [34]

    G. Leoni. A First Course in Sobolev Spaces . Graduate studies in mathematics. American Mathematical Society, 2009

    2009

  28. [35]

    Lorenz and S

    C. Lorenz and S. K¨ oster. Multiscale architecture: Mechanics of composite cytoskeletal net- works. Biophysics Reviews, 3(3), 2022

    2022

  29. [36]

    Maskarinec, C

    S. Maskarinec, C. Franck, D. Tirrell, and G. Ravichandran. Quantifying cellular traction forces in three dimensions. Proceedings of the National Academy of Sciences , 106(52):22108–22113, 2009

    2009

  30. [38]

    C. B. Morrey. Regularity theorems for the solutions of general elliptic systems and boundary value problems, pages 209–286. Springer Berlin Heidelberg, Berlin, Heidelberg, 1966

    1966

  31. [39]

    J. L. Mueller and S. Siltanen. Linear and Nonlinear Inverse Problems with Practical Applica- tions. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2012

    2012

  32. [40]

    M. R. Ng, A. Besser, J. S. Brugge, and G. Danuser. Mapping the dynamics of force transduction at cell-cell junctions of epithelial clusters. eLife, 3:e03282, dec 2014

    2014

  33. [41]

    K. B. Petersen and M. S. Pedersen. The Matrix Cookbook. Technical University of Denmark, 2012

    2012

  34. [42]

    S. V. Plotnikov, B. Sabass, U.S. Schwarz, and C. M. Waterman. Chapter 20 - High-Resolution Traction Force Microscopy. In Jennifer C. Waters and Torsten Wittman, editors, Quantitative Imaging in Cell Biology , volume 123 of Methods in Cell Biology , pages 367–394. Academic Press, 2014

    2014

  35. [43]

    Sabass, M

    B. Sabass, M. L. Gardel, C. M. Waterman, and U.S. Schwarz. High Resolution Traction Force Microscopy Based on Experimental and Computational Advances. Biophysical Journal, 94(1):207–220, 2008

    2008

  36. [44]

    J. A. Sanz-Herrera, J. Barrasa-Fano, M. C´ ondor, and H. Van Oosterwyck. Inverse method based on 3D nonlinear physically constrained minimisation in the framework of traction force microscopy. Soft Matter , 17:10210–10222, 2021

    2021

  37. [45]

    Sch¨ oberl

    J. Sch¨ oberl. C++11 Implementation of Finite Elements in NGSolve (ASC Report 30/2014). Institute of Analysis and Scientific Computing, TU Wien , page 1–23, 2014

    2014

  38. [46]

    U.S. Schwarz. Mechanobiology by the numbers: a close relationship between biology and 27 physics. Nature Reviews Molecular Cell Biology , 18(12):711–712, 2017

    2017

  39. [47]

    Schwarz, N.Q

    U.S. Schwarz, N.Q. Balaban, D. Riveline, A. Bershadsky, B. Geiger, and S.A. Safran. Calcu- lation of forces at focal adhesions from elastic substrate data: The effect of localized force and the need for regularization. Biophysical Journal, 83(3):1380–1394, 2002

    2002

  40. [48]

    Schwarz and J

    U.S. Schwarz and J. Soin´ e. Traction force microscopy on soft elastic substrates: A guide to recent computational advances. Biochimica et Biophysica Acta (BBA) - Molecular Cell Research, 1853(11, Part B):3095–3104, 2015. Mechanobiology

    2015

  41. [49]

    V.I. Smirnov. Chapter IV - metric and normed spaces. In V.I. Smirnov, editor, A Course of Higher Mathematics , volume 62 of International Series of Monographs on Pure and Applied Mathematics, pages 257–366. Pergamon, 1964

    1964

  42. [50]

    Soin´ e.Reconstruction and Simulation of Cellular Traction Forces

    J. Soin´ e.Reconstruction and Simulation of Cellular Traction Forces . PhD thesis, Heidelberg University, 2014

    2014

  43. [51]

    Soin´ e, C

    J. Soin´ e, C. Brand, J. Stricker, P. Oakes, M. Gardel, and U.S. Schwarz. Model-based Traction Force Microscopy Reveals Differential Tension in Cellular Actin Bundles. PLoS computational biology, 11:e1004076, 03 2015

    2015

  44. [52]

    Style, R

    R. Style, R. Boltyanskiy, G. German, C. Hyland, C. MacMinn, A. Mertz, L. Wilen, and E. Dufresne. Traction force microscopy in physics and biology. Soft matter , 10, 04 2014

    2014

  45. [53]

    Suchocki and S

    C. Suchocki and S. Jemio lo. Polyconvex hyperelastic modeling of rubberlike materials. Journal of the Brazilian Society of Mechanical Sciences and Engineering , 43, 07 2021

    2021

  46. [54]

    Tambe, U

    D. Tambe, U. Croutelle, X. Trepat, C. Park, J. H. Kim, E. Millet, J. Butler, and J. Fredberg. Monolayer Stress Microscopy: Limitations, Artifacts, and Accuracy of Recovered Intercellular Stresses. PloS one , 8:e55172, 02 2013

    2013

  47. [55]

    Tambe, C

    D. Tambe, C. Hardin, J. Fredberg, and X. Trepat. Collective cell guidance by cooperative intercellular forces. Nature Precedings, 12 2010

    2010

  48. [56]

    L. Tartar. The Equivalence Lemma; Compact Embeddings , pages 53–57. Springer Berlin Heidelberg, Berlin, Heidelberg, 2007

    2007

  49. [57]

    Toyjanova, E

    J. Toyjanova, E. Bar-Kochba, C. L´ opez-Fagundo, J. Reichner, D. Hoffman-Kim, and C. Franck. High Resolution, Large Deformation 3D Traction Force Microscopy. PLOS ONE , 9:1–12, 04 2014

    2014

  50. [58]

    T. Valent. Boundary Value Problems of Finite Elasticity: Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data . Springer Tracts in Natural Philosophy. Springer New York, 1988

    1988

  51. [59]

    Zancla, P

    A. Zancla, P. Mozetic, M. Orsini, G. Forte, and A. Rainer. A primer to traction force mi- croscopy. Journal of Biological Chemistry , 298(5), 2022

    2022

  52. [60]

    E. H. Zeidler. Nonlinear Functional Analysis and its Applications: IV: Applications to Math- ematical Physics. Springer New York, 1997

    1997

  53. [61]

    Zelen´ a, J

    A. Zelen´ a, J. Blumberg, D. Probst, R. Gerasimait˙ e, G. Lukinavi˘ cius, U.S. Schwarz, and S. K¨ oster. Force generation in human blood platelets by filamentous actomyosin structures. Biophysical Journal, 122(16):3340–3353, 2023. 28

    2023

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.