A new asymptotically correct contact model has been developed for conical tip based atomic force microscopy (AFM) nanoindentation. This new model provides both elastic and nonspecific adhesion properties of cells and soft gels by taking sample thickness at the point of indentation and its depth of indentation into consideration. The bottom substrate effect (BSE) is the most common source of error in the study of “AFM force maps” of the cellular sample. The present model incorporates an asymptotically correct correction term as a function of depth of indentation to eliminate the substrate effect in the analysis. Later, the model is extended to analyze the unloading portion of the indentation curve to extract the stiffness and adhesive properties simultaneously. A comparative study of the estimated material properties using other established contact models shows that the provided corrections effectively curb the errors coming from infinite thickness assumption. Nonspecific adhesive nature of a cell is represented in terms of adhesion parameter (γa) based on the “work of adhesion,” this is an alternative to the peak value of tip–sample attractive (negative) force commonly used as representative adhesion measurement. The simple analytical expression of the model can help in estimating more realistic and accurate biomechanical properties of cells from atomic force microscopy based indentation technique.

References

References
1.
Suresh
,
S.
,
2007
, “
Biomechanics and Biophysics of Cancer Cells
,”
Acta Biomater.
,
3
(
4
), pp.
413
438
.
2.
Li
,
Q. S.
,
Lee
,
G. Y.
,
Ong
,
C. N.
, and
Lim
,
C. T.
,
2008
, “
AFM Indentation Study of Breast Cancer Cells
,”
Biochem. Biophys. Res. Commun.
,
374
(
4
), pp.
609
613
.
3.
Palmieri
,
V.
,
Lucchetti
,
D.
,
Maiorana
,
A.
,
Papi
,
M.
,
Maulucci
,
G.
,
Calapa
,
F.
,
Ciasca
,
G.
,
Giordano
,
R.
,
Sgambato
,
A.
, and
De Spirito
,
M.
,
2015
, “
Mechanical and Structural Comparison Between Primary Tumor and Lymph Node Metastasis Cells in Colorectal Cancer
,”
Soft Matter
,
11
(
8
), pp.
5719
5726
.
4.
Zhou
,
Z.
,
Zheng
,
C.
,
Li
,
S.
,
Zhou
,
X.
,
Liu
,
Z.
,
He
,
Q.
,
Zhang
,
N.
,
Ngan
,
A.
,
Tang
,
B.
, and
Wang
,
A.
,
2013
, “
AFM Nanoindentation Detection of the Elastic Modulus of Tongue Squamous Carcinoma Cells With Different Metastatic Potentials
,”
Nanomed. Nanotechnol. Biol. Med.
,
9
(
7
), pp.
864
874
.
5.
Cross
,
S. E.
,
Jin
,
Y.-S.
,
Rao
,
J.
, and
Gimzewski
,
J. K.
,
2009
, “
Applicability of AFM in Cancer Detection
,”
Nat. Nanotechnol.
,
4
(
2
), pp.
72
73
.
6.
Cross
,
S. E.
,
Jin
,
Y.-S.
,
Rao
,
J.
,
Tondre
,
J.
,
Wong
,
R.
, and
Gimzewski
,
J. K.
,
2008
, “
AFM-Based Analysis of Human Metastatic Cancer Cells
,”
Nanotechnology
,
19
(
38
), p.
384003
.
7.
Hung
,
M.-S.
, and
Tsai
,
M.-F.
,
2015
, “
Investigating the Influence of Anti-Cancer Drugs on the Mechanics of Cells Using AFM
,”
BioNanoScience
,
5
(
3
), pp.
156
161
.
8.
Efremov
,
Y. M.
,
Dokrunova
,
A. A.
,
Efremenko
,
A. V.
,
Kirpichnikov
,
M. P.
,
Shitan
,
K. V.
, and
Sokolova
,
O. S.
,
2015
, “
Distinct Impact of Targeted Actin Cytoskeleton Reorganization on Mechanical Properties of Normal and Malignant Cells
,”
Biochim. Biophys. Acta
,
1853
(
11 Pt. B
), pp.
3117
3125
.
9.
Morton
,
K. C.
, and
Baker
,
L. A.
,
2014
, “
Atomic Force Microscopy-Based Bioanalysis for the Study of Disease
,”
Anal. Methods
,
6
(
14
), pp.
4932
4955
.
10.
Banguy
,
X.
,
Suarez
,
F.
,
Argaw
,
A.
,
Bouchard
,
J.-F.
,
Hildgen
,
P.
, and
Giasson
,
S.
,
2009
, “
Effect of Mechanical Properties of Hydrogel Nanoparticles on Macrophage Cell Uptake
,”
Soft Matter
,
5
(
20
), pp.
3984
3991
.
11.
Vasir
,
J. K.
, and
Labhasetwar
,
V.
,
2008
, “
Quantification of the Force of Nanoparticle-Cell Membrane Interactions and Its influence on Intracellular Trafficking of Nanoparticles
,”
Biomaterials
,
29
(
31
), pp.
4244
4252
.
12.
Haga
,
H.
,
Sasaki
,
S.
,
Kawabata
,
K.
,
Ito
,
E.
,
Ushiki
,
T.
, and
Sambongi
,
T.
,
2000
, “
Elasticity Mapping of Living Fibroblasts by AFM and Immunofluorescence Observation of the Cytoskeleton
,”
Ultramicroscopy
,
82
(1–4), pp.
253
258
.
13.
Wagh
,
A. A.
,
Roan
,
E.
,
Chapman
,
K. E.
,
Desai
,
L. P.
,
Rendon
,
D. A.
,
Eckstein
,
E. C.
, and
Waters
,
C. M.
,
2008
, “
Localized Elasticity Measured in Epithelial Cells Migrating at a Wound Edge Using Atomic Force Microscopy
,”
Lung Cell. Mol. Physiol.
,
295
(
1
), pp.
L54
L60
.
14.
Hansen
,
J. C.
,
Lim
,
J. Y.
,
Xu
,
L.
,
Siedleckia
,
C. A.
,
Mauger
,
D. T.
, and
Donahu
,
H. J.
,
2007
, “
Effect of Surface Nanoscale Topography on Elastic Modulus of Individual Osteoblastic Cells as Determined by Atomic Force Microscopy
,”
J. Biomech.
,
40
(
13
), pp.
2865
2871
.
15.
Wang
,
B.
,
Guo
,
P.
, and
Auguste
,
D. T.
,
2015
, “
Mapping the CXCR4 Receptor on Breast Cancer Cells
,”
Biomaterials
,
57
, pp.
161
168
.
16.
Almqvist
,
N.
,
Bhatia
,
R.
,
Primbs
,
G.
,
Desai
,
N.
,
Banerjee
,
S.
, and
Lal
,
R.
,
2004
, “
Elasticity and Adhesion Force Mapping Reveals Real-Time Clustering of Growth Factor Receptors and Associated Changes in Local Cellular Rheological Properties
,”
Biophys. J.
,
86
(
3
), pp.
1753
1762
.
17.
Zammaretti
,
P. S.
, and
Ubbink
,
J.
,
2003
, “
Imaging of Lactic Acid Bacteria With AFM—Elasticity and Adhesion Maps and Their Relationship to Biological and Structural Data
,”
Ultramicroscopy
,
97
(1–4), pp.
199
208
.
18.
Sirghi
,
L.
,
Ponti
,
J.
,
Broggi
,
F.
, and
Rossi
,
F.
,
2008
, “
Probing Elasticity and Adhesion of Live Cells by Atomic Force Microscopy Indentation
,”
Eur. Biophys. J.
,
37
(
6
), pp.
935
945
.
19.
Johnson
,
K. L.
,
2003
,
Contact Mechanics
,
Cambridge University Press
,
Cambridge, UK
, Chap. 4.
20.
Sneddon
,
I. N.
,
1965
, “
The Relation Between Load and Penetration in the Axisymmetric Boussinesq Problem for a Punch of Arbitrary Profile
,”
Int. J. Eng. Sci.
,
3
(
1
), pp.
47
57
.
21.
Sun
,
Y.
,
Akhremitchev
,
B.
, and
Walker
,
G. C.
,
2004
, “
Using the Adhesive Interaction Between Atomic Force Microscopy Tips and Polymer Surfaces to Measure the Elastic Modulus of Compliant Samples
,”
Langmuir
,
20
(
14
), pp.
5837
5845
.
22.
Sen
,
S.
,
Subramanian
,
S.
, and
Discher
,
D.
,
2005
, “
Indentation and Adhesive Probing of a Cell Membrane With AFM: Theoretical Model and Experiments
,”
J. Biophys.
,
89
(
5
), pp.
3203
3213
.
23.
Johnson
,
K. L.
,
Kendall
,
K.
, and
Roberts
,
A. D.
,
1971
, “
Surface Energy and the Contact of Elastic Solids
,”
Proc. R. Soc. London
,
324
(
1558
), pp.
301
313
.
24.
Sirghi
,
L.
,
2010
, “
Atomic Force Microscopy Indentation of Living Cells
,”
Microscopy: Science, Technology, Applications and Education
, Formatex Research Center, Badajoz, Spain, pp.
433
440
.
25.
Sirghi
,
L.
, and
Rossi
,
F.
,
2006
, “
Adhesion and Elasticity in Nanoscale Indentation
,”
Appl. Phys. Lett.
,
89
(
24
), p.
243118
.
26.
Costa
,
K. D.
, and
Yin
,
F. C. P.
,
1999
, “
Analysis of Indentation: Implications for Measuring Mechanical Properties With Atomic Force Microscopy
,”
ASME J. Biomech. Eng.
,
121
(
5
), pp.
462
471
.
27.
Xu
,
W.
,
Mezencev
,
R.
,
Kim
,
B.
,
Wang
,
L.
,
McDonald
,
J.
, and
Sulchek
,
T.
,
2012
, “
Cell Stiffness Is a Biomarker of the Metastatic Potential of Ovarian Cancer Cells
,”
PLoS One
,
7
(
10
), p.
e46609
.
28.
Crick
,
S. L.
, and
Yin
,
F. C.-P.
,
2007
, “
Assessing Micromechanical Properties of Cells With Atomic Force Microscopy: Importance of the Contact Point
,”
Biomech. Modell. Mechanobiol.
,
6
(
3
), pp.
199
210
.
29.
Kim
,
Y.
,
Kim
,
M.
,
Shin
,
J. H.
, and
Kim
,
L.
,
2011
, “
Characterization of Cellular Elastic Modulus Using Structure Based Double Layer
,”
Med. Biol. Eng. Comput.
,
49
(
4
), pp.
453
462
.
30.
Dimitriadis
,
E. K.
,
Horkay
,
F.
,
Maresca
,
J.
,
Kachar
,
B.
, and
Chadwick
,
R. S.
,
2002
, “
Determination of Elastic Moduli of Thin Layers of Soft Material Using the Atomic Force Microscope
,”
J. Biophys.
,
82
(
5
), pp.
2798
2810
.
31.
Mahaffy
,
R. E.
,
Park
,
S.
,
Ferde
,
E.
,
Kas
,
J.
, and
Shih
,
C. K.
,
2004
, “
Quantitative Analysis of the Viscoelastic Properties of Thin Regions of Fibroblasts Using Atomic Force Microscopy
,”
Biophys. J.
,
86
(
3
), pp.
1777
1793
.
32.
Santos
,
J. A. C.
,
Rebelo
,
L. M.
,
Araujo
,
A. C.
,
Barros
,
E. B.
, and
De Sousa
,
J. S.
,
2012
, “
Thickness-Corrected Model for Nanoindentation of Thin Films With Conical Indenters
,”
Soft Matter
,
8
(
16
), pp.
4441
4148
.
33.
Gavara
,
N.
, and
Chadwick
,
R. S.
,
2012
, “
Determination of the Elastic Moduli of Thin Samples and Adherent Cells Using Conical Atomic Force Microscope Tips
,”
Nat. Nanotechnol.
,
7
(
11
), pp.
733
736
.
34.
Lin
,
D. C.
, and
Horkay
,
F.
,
2008
, “
Nanomechanics of Polymer Gels and Biological Tissues: A Critical Review of Analytical Approaches in the Hertzian Regime and Beyond
,”
Soft Matter
,
4
(
4
), pp.
669
68
2
35.
Betti
,
E.
,
1872
,
IlnuovoCimento (Societaitaliana di Fisica)
, Italian Physical Society, Bologna, Italy.
36.
Lord Rayleigh,
1873
, “London Mathematics Society Proceedings,” The London Mathematical Society, London.
37.
Landau
,
L. D.
, and
Lifshitz
,
E. M.
,
1975
,
Theory of Elasticity
,
Pergamon Press
,
Oxford, UK
, Chap. 1.
38.
Shield
,
R. T.
,
1967
, “
Load Displacement Relations for Elastic Bodies
,”
Z. Angew. Math. Phys.
,
18
(
5
), pp.
682
693
.
39.
Israelachivili
,
J. N.
,
1992
,
Intermolecular and Surface Forces
,
3rd ed.
,
Academic Press
, Santa Barbara, CA, Chap. 10.
40.
Tse
,
J. R.
, and
Engler
,
A. J.
,
2010
, “
Preparation of Hydrogel Substrates With Tuneable Mechanical Properties
,”
Current Protocols in Cell Biology
, Wiley, Hoboken, NJ, Chap. 10.
41.
Pelham
,
R. J.
, and
Wang
,
Y.
,
1997
, “
Cell Locomotion and Focal Adhesion Are Regulated by Substrate Flexibility
,”
Proc. Natl. Acad. Sci. U.S.A.
,
94
(
25
), pp.
13661
13665
.
42.
Schaus
,
Y. A.
,
Okes
,
P. W.
,
Sticker
,
J.
,
Winter
,
S. P.
, and
Gardel
,
M. L.
,
2010
, “
Preparation of Complaint Matrices for Quantifying Cellular Contraction
,”
JOVE
,
46
, p.
2173
.
43.
Hutter
,
J. L.
, and
Bechhoefer
,
1993
, “
Calibration of Atomic Force Microscope Tips
,”
Rev. Sci. Instrum.
,
64
(
7
), pp.
1868
1873
.
44.
Darling
,
E. M.
,
Zauscher
,
S.
,
Block
,
J. A.
, and
Guilak
,
F.
,
2007
, “
A Thin-Layer Model for Viscoelastic, Stress-Relaxation Testing of Cells Using Atomic Force Microscopy: Do Cell Properties Reflect Metastatic Potential?
,”
J. Biophys.
,
92
(
5
), pp.
1784
1791
.
45.
MathWorks, 2014, “
MATLAB and Statistics Toolbox Release 2014
,” The MathWorks, Inc., Natick, MA.
46.
Gunning
,
A. P.
,
Chambers
,
S.
,
Pin
,
C.
,
Man
,
A. L.
,
Morris
,
V. J.
, and
Nicoletti
,
C.
,
2008
, “
Mapping Specific Adhesive Interaction on Living Human Intestinal Epithelial Cells With Atomic Force Microscopy
,”
FASEB J.
,
22
(
7
), pp.
2331
2339
.
47.
Zhu
,
C.
,
2000
, “
Kinetics and Mechanics of Cell Adhesion
,”
J. Biomech.
,
33
(
1
), pp.
23
33
.
48.
Zhu
,
A. P.
,
Fang
,
N.
,
Chan-Park
,
M. B.
, and
Chan
,
V.
,
2006
, “
Adhesion Contact Dynamics of 3T3 fibroblasts on Poly (Lactide-co-Glycolide Acid) Surface Modified by Photochemical Immobilization of Biomacromolecules
,”
Biomaterials
,
12
(12), pp.
2566
2576
.
49.
Mathur
,
A. B.
,
Collinsworth
,
A. M.
,
Reichert
,
W. M.
,
Kraus
,
W. E.
, and
Truskey
,
G. A.
,
2001
, “
Endothelial, Cardiac Muscle and Skeletal Muscle Exhibit Different Viscous and Elastic Properties as Determined by Atomic Force Microscopy
,”
J. Biomech.
,
34
(
12
), pp.
1545
1553
.
50.
Wolfram Research,
2012
, “
Mathematica (Programming Language), Version 9.0
,” Wolfram Research Inc., Champaign, IL.
You do not currently have access to this content.