Mechanical mapping of thin elastic films and living cells with spherical tip atomic force microscopy probes at large indentations

Annalisa Cal\`o; Beatriz Cantero; Eleni Dalaka; Gabriel Gomila; Giuseppe Battaglia; Jordi Comelles; Lara Aiassa; Loris Rizzello; Mauricio Cano; Sophie Strawbridge

REVIEW 3 major objections 5 minor 3 references

Mechanical mapping of thin elastic films and living cells with spherical tip atomic force microscopy probes at large indentations

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A closed-form model now fits spherical-tip AFM force curves on thin films at large indentations, revealing that PAA hydrogels stiffen below ~15 µm thickness.

desk verdict Useful semi-empirical spherical-tip BEC model for AFM at large indentations; the PAA stiffening claim is plausible but needs an out-of-sample FE check. read the letter →

arxiv 2607.22167 v1 pith:B2QMM3AM submitted 2026-07-24 cond-mat.mtrl-sci cond-mat.soft

Gabriel Gomila , Mauricio Cano , Beatriz Cantero , Sophie Strawbridge , Lara Aiassa , Loris Rizzello , Giuseppe Battaglia , Eleni Dalaka

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Jordi Comelles Annalisa Calò

This is my paper · ORCID

classification cond-mat.mtrl-scicond-mat.soft MSC 74B0574M15 PACS 07.79.Lh68.35.Gy87.64.Dz

keywords atomicforcemicroscopythinfilmssphericaltipbottomeffectYoung'smodulusPAAhydrogelsmacrophagescontactmechanics

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 presents an analytical force-indentation model for spherical-tip atomic force microscopy on elastic thin films bonded to rigid substrates, valid at large indentations where previous models fail. The model extends an existing series solution to the regime where the contact radius exceeds the film thickness, using numerical contact-mechanics data to fit the coefficients. The authors use this model to show that the intrinsic Young's modulus of PAA hydrogels increases sharply below a critical thickness of ~15 µm, reaching ~80 kPa near 6 µm, a variation they argue is not a bottom-stiffness artifact. They also apply the model to living macrophage cells, obtaining a Young's modulus of about 200 Pa.

What carries the argument

The central object is the closed-form force-indentation expression Eq. (4) (explicit for ν=0.5 as Eq. (5)), which normalizes the force by Sneddon's semi-infinite sphere model and multiplies it by polynomial corrections in (R/h) and (δ/R). Its coefficients a_n(ν) are determined by least-squares fitting to numerical solutions of the contact mechanics integral equation (Perriot-Barthel formulation). This functional form is assumed to hold for all a/h, extending the Taylor-derived form valid only for a/h<1.

What would settle it

Compare the model's prediction for a thin film with known elastic modulus (e.g., a silicone or gelatin film) against finite element simulations that use an independently measured bulk modulus, and check whether the fitted modulus remains constant for thicknesses from 2 µm to 20 µm. A systematic deviation would indicate residual model error.

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Extended reading notes

Core claim

The paper claims that a simple closed-form expression, Eq. (5), accurately reproduces numerically computed force-indentation curves for a spherical indenter on a thin elastic film over the entire range of indentations and thicknesses, including the previously inaccessible regime a/h>1, with relative error below 1.3%. This expression is a phenomenological extension of the Dhaliwal-Rau series solution, with coefficients fitted to numerical data. Using this model, the paper demonstrates that the intrinsic Young's modulus of PAA hydrogel thin films is thickness-dependent below about 15 µm, and that standard models (Sneddon, Hertz, paraboloid with BEC) either overestimate or underestimate the mod

Load-bearing premise

The model's functional form, taken from a small-a/h Taylor expansion, is assumed to remain valid for all a/h, with only coefficients adjusted to numerics; if this form is not flexible enough or the numerical benchmark is biased, the extracted thickness-dependent PAA modulus could be an artifact.

Editorial extensions

If this is right

Spherical-tip AFM users can now analyze force curves at indentations up to min(h,R) without resorting to finite element simulations, obtaining thickness-corrected Young's moduli.
The model provides physically meaningful contact points, correcting topography maps that are distorted when using incorrect models with free contact-point fitting.
The reported thickness-dependent intrinsic modulus of PAA hydrogels implies that mechanobiology experiments using thin hydrogel substrates may need reinterpretation.
For soft living cells like macrophages, the model gives lower Young's moduli than Sneddon's model (by ~100%) and slightly higher than paraboloid BEC models (by ~25%).
The model can be extended to other indenter geometries and compliant substrates by following the same phenomenological fitting procedure.

Reading between the lines

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

If the intrinsic stiffening of thin PAA films is real, it suggests that hydrogel thin films used in cell culture may present a stiffer mechanical environment than bulk measurements indicate, potentially confounding studies of cell mechanosensing on soft substrates.
The method of deriving the functional form from a truncated series and then fitting coefficients to numerics could be applied to other tip geometries (e.g., capped cones or truncated pyramids) to create similar closed-form corrections for large indentations.
A testable extension: measuring PAA films of thickness below 15 µm with a different technique (e.g., indentation with a flat punch or microsphere gravimetry) should reproduce the same intrinsic thickness-dependent modulus if the claim is correct.
The exponential fit E(h) = E0 + α exp(-h/h0) with h0=1.5 µm is purely phenomenological; identifying the physical mechanism (e.g., altered swelling, porosity, or interface effects) would require direct structural measurements.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my take on arXiv:2607.22167. It gives the community something genuinely useful: a closed-form force-indentation expression for a spherical AFM tip on a thin elastic film that covers the a/h>1 regime and indentations up to δ~min(h,R). Dhaliwal-Rau, Sneddon, and paraboloid-with-BEC models all break down there, so this fills a real gap. The authors also provide the Mathematica notebooks, simulation data, and raw experimental data, which makes the work easy to check.

The main weakness is exactly what the stress-test note says: the '<1.3% error' against numerics is a training residual. The coefficients in Eq. (5) are least-squares fitted to those same numerical curves, and the fit is then quoted as validation. The paper is honest about this—Appendix C calls the fitted coefficients 'phenomenological parameters' and the derivation is explicitly labelled a phenomenological strategy—but the abstract says 'analytical model,' which oversells the a/h>1 branch. That branch is an assumed functional form taken from the small-a/h expansion, not a derivation.

The bigger question is the experimental claim that PAA's intrinsic modulus rises sharply below ~15 µm. The experimental fits themselves are excellent (R²>0.999), and the 100%/-25% shifts relative to Sneddon and paraboloid models match the numerics. But E is a free parameter, so a small thickness-dependent bias in Eq. (5) would be absorbed into E and would look exactly like an intrinsic stiffening. The authors partly answer this by showing the observed increase is much larger than the bottom-effect correction and by reproducing it on uniform-thickness samples—but those samples used a slightly different formulation. I don't think this is fatal; it's a caution on interpretation.

Bottom line: worth a serious referee. The model is plausible, well-documented, and tested on independent experiments. Read this if you routinely fit AFM force curves on cells or thin gels; the cell-mechanics crowd will care about the macrophage numbers. What would push me to full accept: an out-of-sample finite-element calculation not used in the fit, plus a sentence in the abstract that the a/h>1 branch is semi-empirical. I'd send it to review, conditional.

Referee Report

3 major / 5 minor

Summary. The paper proposes an explicit closed-form force-indentation model, Eq. (4)/(5), for spherical-tip AFM probes indenting a homogeneous, isotropic, linear elastic thin film bonded to a rigid substrate, covering the entire indentation range including large indentations and a/h>1. The functional form is obtained from a Taylor expansion of the Dhaliwal–Rau analytical solution valid for a/h<1, and the coefficients a1..a9 are then fitted to numerical solutions of the contact integral equation. The model is applied to polyacrylamide (PAA) hydrogel films of controlled thickness and to living macrophages. The main material claim is that PAA films thinner than about 15 μm exhibit an intrinsic increase in Young's modulus, up to about 80 kPa near 6 μm, and that Sneddon/paraboloid models overestimate or underestimate moduli by about 100% and −25%, respectively.

Significance. The paper addresses a real and recognized need in AFM-based mechanobiology: no accurate, easy-to-use model exists for spherical probes on thin elastic films at large indentations, where paraboloid-based BEC models fail. If the proposed model is genuinely predictive outside its fitting range, it would be a useful practical tool. The manuscript is unusually open in sharing data and code: the numerical notebooks, raw experimental data, and coefficient tables for multiple Poisson ratios are provided. The PAA and macrophage force-curve fits are of very high quality (R²>0.999 and ~0.994, respectively), and the comparison with existing models is carefully worked out. The central numerical validation, however, is in-sample: the coefficients were least-squares fitted to the very same numerical data used to report the <1.3% error, so the claimed agreement does not by itself establish predictive accuracy. Because the paper's most impactful conclusion—an intrinsic thickness-dependent PAA modulus—depends on the model being unbiased across h/R, this issue is load-bearing and needs to be addressed with a genuine out-of-sample test before the claim can be fully accepted.

major comments (3)

[§2 and Appendix C, Eq. (C3), Fig. 3] The statement that Eq. (5) reproduces numerical data with relative error below 1.3% is a training residual, not an independent validation. The coefficients a1..a9 in Eq. (C3) are determined by least-squares fitting Eq. (4) to the same numerical dataset later used to compute the error. This is particularly problematic for the a/h>1 branch, where the functional form is assumed, not derived. I recommend (i) rephrasing the claim as 'interpolates the numerical data' rather than 'agrees with numerical solutions', and (ii) performing an out-of-sample test: for example, fit the coefficients on a subset of h/R values and test on the withheld ones, or compare with independent finite-element/numerical solutions from the literature, e.g., Refs. [23] and [25]. Without such a test, the possibility of a systematic model bias that grows as the film thins remains open.
[§5 and Fig. 6] The intrinsic PAA stiffening conclusion is vulnerable to model bias absorption. Since Young's modulus E is a free fit parameter, any small systematic error in Eq. (5) that depends on h/R will be absorbed into E and can generate an apparent thickness dependence of the kind reported. The comparison in Fig. 6c to the bottom-effect signatures of Sneddon and paraboloid models does not exclude model bias—it only shows that the observed trend is not the known signature of those simpler models. In addition, the confirmation used uniformly thick PAA samples prepared with a slightly different chemical formulation (6.2% acrylamide / 0.044% bis-acrylamide vs. 7.5% / 0.05% for the variable thickness film), weakening the claim of a chemically independent validation. I suggest one or more of the following: (i) validate Eq. (5) on an elastomer with a known thickness-independent modulus over the same h/R
[§6, §7 and Fig. 7] The macrophage measurements are described as 'experimental validation' of the model, but living macrophages are not homogeneous, isotropic, linear elastic thin films perfectly bonded to a rigid substrate. The model's assumptions are violated by cellular heterogeneity, finite lateral size, and a basal boundary that is not a continuous bonded rigid interface. The high R² of the fits (0.994) does not constitute an independent validation because E and the contact point are free parameters and the model is flexible enough to fit many curves. I recommend relabeling the cell experiments as an illustrative application rather than a validation, and explicitly listing the assumptions that are likely violated for cells.

minor comments (5)

[Throughout] Many equations (e.g., Eq. (4), Eq. (5), Eq. (B8), Eq. (B9)) appear garbled in the manuscript text, with missing operators and mismatched parentheses. The equations must be typeset cleanly before publication.
[§3 and Fig. 4] The text refers to 'Hermanowicz's paraboloid with BEC, Eq. (A2)', but Eq. (A2) is Garcia et al.'s model; Hermanowicz's model is Eq. (A3). This inconsistency appears in several places, including the caption of Fig. 4 and the Methods section.
[§5, Eq. (6)] The fitted parameters of Eq. (6) are inconsistent with the experimental values: with E_PAA,0 = 3.4 kPa, α = 2.3±0.2 GPa, and h0 = 1.5 μm, the predicted value at h = 6 μm is on the order of tens of MPa, not ~80 kPa. This suggests either a typo in the reported α or a misspecified equation. Please clarify the units and the actual fitted expression.
[§5 and Fig. 6] The symbols in Fig. 6 for variable-thickness and uniform-thickness samples are said to correspond to different chemical formulations, but the Methods section gives recipes with different acrylamide concentrations. The text states these have 'similar ratio of monomer and crosslinker', but the ratios differ (7.5/0.05 = 150 vs. 6.2/0.044 ≈ 141). Please justify that these are expected to have the same bulk modulus.
[§4 and Methods] The description of the force-curve fitting says the contact point is left as a free parameter. The paper correctly shows that free contact point can mask model inadequacy, but the authors do not report the contact-point values obtained with their model against the independently estimated breakpoint except in histograms. A direct comparison for representative curves would be useful.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed <1.3% agreement with numerical contact solutions is an in-sample fit (coefficients a1..a9 fitted to those same data), so the numerical validation is partially circular; external PAA/macrophage experiments and the thick-film Sneddon limit give the central model independent content.

fitted input called prediction [Appendix C (Eq. C3) and Section 2 (Eq. 5, Fig. 3)]
"If instead the coefficients in Eq. (4) are treated as phenomenological parameters to be determined by a least squares fit of Eq. (4) to the data calculated numerically, one obtains almost perfect agreement with the numerical data. ... leading to Eq. (5) of the main text, which shows a relative error of less than 1.3% with the numerically calculated data. ... Figure 3 ... shows the predictions of Eq. (5) and compares them with the numerically calculated data (symbols), demonstrating that Eq. (5) reproduces the numerical data with a relative error of less than 1.3%."

The nine coefficients in Eq. (5) are obtained by least-squares fitting Eq. (4) to the same numerically calculated force-indentation curves shown in Fig. 3 (and described in Appendix C). The paper then presents Eq. (5) as 'predicting' or 'reproducing' those curves with <1.3% error. That number is the training residual of the fit, not an out-of-sample test of the assumed functional form. In particular, the a/h>1 branch is not derived from contact mechanics; the Dhaliwal-Rau form is assumed to continue and only its coefficients are fitted, so the <1.3% agreement cannot independently establish that the assumed form captures the physics at a/h>1. The same in-sample character underlies the Section 3 statement that fitted moduli are 'almost identical to the nominal value used in the numerical cal

full rationale

The paper is transparent: it labels the strategy phenomenological, derives the functional form from the a/h<1 Dhaliwal-Rau expansion, and states that Eq. (4)'s coefficients are fitted to numerically calculated data. The circularity is in the validation claim built on that same fit: 'relative error less than 1.3% with the numerically calculated data' is not an independent confirmation of the model, because those data were used to choose the coefficients. This is a genuine fitted-input-called-prediction step and it weakens the numerical-accuracy claim that supports the model's use for thin films. It does not, however, make the whole paper circular. The model reduces to Sneddon in the thick limit, reproduces the independently known a/h<1 Dhaliwal-Rau series when the fitted coefficients are replaced by expansion values, and is tested against external experimental PAA and macrophage force curves (with E as a free parameter). The inferred thickness-dependent PAA modulus is not itself a fit output of Eq. (5)'s coefficient fit, but it inherits the risk that a small systematic bias in the in-sample-fitted model at low h/R is absorbed into the fitted E. No load-bearing self-citation or imported uniqueness theorem was found. Score 6 reflects one central validation that reduces by construction, with independent content remaining outside that step.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central model introduces no new physical entities or forces; it is a semi-empirical closed-form fit to existing contact mechanics. The key burden is the nine fitted coefficients per Poisson ratio plus the assumed functional form for a/h>1; the PAA material claim adds a three-parameter phenomenological exponential fit. No new particles, dimensions, or conserved quantities are postulated.

free parameters (3)

a1..a9 coefficients in Eq. (5) for ν=0.5 = 1.01345, 2.24003, 0.470293, 0.93412, 0.455595, 0.124456, 0.20410, 0.938828, 0.0291373
Nine coefficients are least-squares fitted to numerically calculated force-indentation curves for a spherical indenter on a thin film (Fig. 3 data); they are not derived for a/h>1. Table C1 gives additional fitted sets for other Poisson ratios.
Phenomenological E(h) parameters: E_PAA,0, α, h0 = E_PAA,0=3.4 kPa; α=2.3 GPa; h0=1.5 µm
Fitted to experimental PAA Young's modulus versus thickness data in Eq. (6): E(h)=E0+α exp(-h/h0). Used to characterize the claimed intrinsic stiffening below ~15 µm.
Per-curve Young's modulus and contact point in force-curve fits = e.g., E_sphere=4.8±0.3 kPa for the PAA region; z_c varies per pixel
Standard free parameters in every AFM force-curve fit. They are fitted inversion outputs rather than model constants, but they determine the extracted material maps and topography.

assumptions (5)

domain assumption Film is homogeneous, isotropic, linear elastic, laterally infinite, bottom-bonded to a rigid substrate, with frictionless non-adhesive contact.
Used throughout the contact integral equation and model; PAA and living cells only approximately satisfy these conditions, as acknowledged in Section 7.
domain assumption Poisson's ratio is known (ν=0.5 used for PAA and macrophages; Table C1 for other values).
The coefficients a_n(ν) depend on Poisson's ratio. Incompressibility is plausible for hydrogels and cells but is not independently verified for macrophages.
standard math The Perriot-Barthel contact-mechanics integral equation and its numerical solution are an accurate benchmark.
All 'exact' numerical data in Figs. 2-3 and all coefficient fitting rely on this numerical solver; no independent finite-element validation is presented in this paper.
domain assumption Variable-thickness and uniform-thickness PAA samples with slightly different preparation protocols have the same intrinsic modulus for the same thickness and composition.
Section 5 combines two sample sets to conclude intrinsic E(h); the protocols differ in monomer/crosslinker percentages and crosslinking time but are asserted to have the same nominal bulk modulus.
standard math Sneddon's sphere-on-half-space solution is exact and is the correct thick-film limit of the model.
Used to normalize Eq. (4), to define F_Sned in Eq. (2), and to justify the h/R>50 limit.

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Pith. "Pith review of Mechanical mapping of thin elastic films and living cells with spherical tip atomic force microscopy probes at large indentations." pith.science (2026). https://pith.science/paper/B2QMM3AM

@misc{pith2026260722167,
  author       = {Pith},
  title        = {Pith review of: Mechanical mapping of thin elastic films and living cells with spherical tip atomic force microscopy probes at large indentations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B2QMM3AM}},
  note         = {Machine review of arXiv:2607.22167}
}

read the original abstract

An analytical model to quantify large indentation force curves acquired on elastic thin films and living cells with spherical tip Atomic Force Microscopy (AFM) probes is presented. The model accounts for the bottom effect in the whole indentation range and overcomes the limitations of Sneddon's and Hertz's contact models, which are valid for semi-infinite thick samples, and of paraboloid tip models with bottom effect correction (BEC) that are applicable to spherical tips only for relatively small indentations. The model is experimentally validated with force volume measurements on polyacrylamide (PAA) hydrogel thin films, where an excellent agreement is obtained. The accurate correction of the bottom effect demonstrates that the intrinsic Young's modulus of PAA thin films increases for thickness below a critical value (~15 um). The model also shows excellent agreement with force curves acquired on live macrophages, providing accurate Young's modulus values for these very soft cells (E~200 Pa). Young's modulus values extracted with the proposed model significantly differ from those obtained from Sneddon's or paraboloid models with BEC, whose values deviate by 100% and -25%, respectively. Results show the potential of the proposed model for analysing force curve measurements with spherical tips at large indentations on thin film elastic materials and living cells at the micro and nanoscale.

Figures

Figures reproduced from arXiv: 2607.22167 by the authors.

**Figure 1.** (a) and (b) Scanning Electron Microscopy images of colloidal and electron beam deposited spherical tip AFM probes, respectively. (c) Schematic representation of a sphere of radius R indenting an elastic thin film of thickness h, Young's modulus E and Poisson's ratio ν bonded to a rigid substrate under the application of a frictionless loading force F. a is the contact radius and δ the indentation. For thin samples a… view at source ↗

**Figure 2.** (black symbols) shows a force-indentation curve numerically calculated for a spherical indenter of radius R, indenting a thin elastic film of Young's modulus E, Poisson's ratio ν=0.5 and thickness h=1.25·R bonded to a rigid substrate for the case of non-adhesive frictionless contact. The numerical results were obtained by solving the corresponding contact mechanics integral equation according to the formulation in R… view at source ↗

**Figure 3.** (symbols) Dimensionless force-indentation curves calculated numerically by solving the contact mechanics integral equation for different ratios of sample thickness to probe radius, h/R, for ν=0.5. (Black dashed lines) Force-indentation curves predicted by the truncated analytical solution of Dhaliwal and Rau [22], Eq(B8). (Colour continuous lines) Forceindentation curves predicted by the analytical model proposed i… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: (Black dashed and continuous lines) Least-squares fitting of (a) Hertz's, (b) Sneddon's and (c) Hermanowicz's paraboloids with BEC tip models to numerically calculated data for a spherical indenter on a thin film (symbols, same data as in [PITH_FULL_IMAGE:figures/full…

**Figure 6.** Figure 6: (a) (Solid symbols) Cumulative representation of the extracted Young's modulus in the six regions probed on the variable thickness PAA film versus the true sample thickness. (c) (b) (a) [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: (a) The AFM topographic image shows a THP1-M0 macrophage living cell with a force setpoint of Fset=7 nN, acquired with a colloidal AFM probe of radius R=5 µm. (b) (Black line) Cross-section topographic profile along the white dashed line in (a). (Red dashed line) True …

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Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Determination of elastic moduli of thin layers of soft material using the atomic force microscope,

E. K. Dimitriadis, F. Horkay, J. Maresca, B. Kachar, and R. S. Chadwick, “Determination of elastic moduli of thin layers of soft material using the atomic force microscope,” Biophys J, vol. 82, no. 5, pp. 2798–2810, 2002, doi: 10.1016/S0006-3495(02)75620-8

work page doi:10.1016/s0006-3495(02)75620-8 2002
[2]

Determination of the Elastic Moduli of a Single Cell Cultured on a Rigid Support by Force Microscopy,

P. D. Garcia and R. Garcia, “Determination of the Elastic Moduli of a Single Cell Cultured on a Rigid Support by Force Microscopy,” Biophys J, vol. 114, no. 12, pp. 2923–2932, Jun. 2018, doi: 10.1016/j.bpj.2018.05.012

work page doi:10.1016/j.bpj.2018.05.012 2018
[3]

Effects of gel thickness on microscopic indentation measurements of gel modulus,

R. Long, M. S. Hall, M. Wu, and C. Y. Hui, “Effects of gel thickness on microscopic indentation measurements of gel modulus,” Biophys J , vol. 101, no. 3, pp. 643–650, Aug. 2011, doi: 10.1016/j.bpj.2011.06.049

work page doi:10.1016/j.bpj.2011.06.049 2011

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Determination of elastic moduli of thin layers of soft material using the atomic force microscope,

E. K. Dimitriadis, F. Horkay, J. Maresca, B. Kachar, and R. S. Chadwick, “Determination of elastic moduli of thin layers of soft material using the atomic force microscope,” Biophys J, vol. 82, no. 5, pp. 2798–2810, 2002, doi: 10.1016/S0006-3495(02)75620-8

work page doi:10.1016/s0006-3495(02)75620-8 2002

[2] [2]

Determination of the Elastic Moduli of a Single Cell Cultured on a Rigid Support by Force Microscopy,

P. D. Garcia and R. Garcia, “Determination of the Elastic Moduli of a Single Cell Cultured on a Rigid Support by Force Microscopy,” Biophys J, vol. 114, no. 12, pp. 2923–2932, Jun. 2018, doi: 10.1016/j.bpj.2018.05.012

work page doi:10.1016/j.bpj.2018.05.012 2018

[3] [3]

Effects of gel thickness on microscopic indentation measurements of gel modulus,

R. Long, M. S. Hall, M. Wu, and C. Y. Hui, “Effects of gel thickness on microscopic indentation measurements of gel modulus,” Biophys J , vol. 101, no. 3, pp. 643–650, Aug. 2011, doi: 10.1016/j.bpj.2011.06.049

work page doi:10.1016/j.bpj.2011.06.049 2011