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Nanoindentation technology has proven to be an effective method to investigate the viscoelastic properties of biological cells. The experimental data obtained by nanoindentation are frequently interpreted by Hertz contact model. However, in order to validate Hertz contact model, some studies assume that cells have infinite thickness which does not necessarily represent the real situation. In this study, a rigorous contact model based upon linear elasticity is developed for the interpretation of indentation tests of flattened cells. The cell, normally bonded to the Petri dish, is initially treated as an elastic layer of finite thickness perfectly fixed to a rigid substrate. The theory of linear elasticity is utilized to solve this contact issue and then the solutions are extended to viscoelastic situation which is regarded as a good indicator for mechanical properties of biological cells. To test the present model, AFM-based creep test has been conducted on living human hepatocellular carcinoma cell (SMMC-7721 cell) and its fullerenol-treated counterpart. The results indicate that the present model could not only describe very well the creep behavior of SMMC-7721 cells, but also curb overestimation of the mechanical properties due to substrate effect.

The measurement of viscoelastic properties of living cells can provide important information about the biomechanical effects of drug treatment, diseases, and aging. To date, a variety of testing techniques have been used to measure the viscoelastic mechanical properties of biological cells, for example, micropipette aspiration [

The indentation of thin layer by spherical indenters has been commonly studied in the literature using either cumbersome numerical calculations or analytical modeling [

In this work, we use linear theory of elasticity to develop a new correction to Sneddon’s solutions [

Consider the axisymmetric contact problem of a rigid conic tip on an elastic layer as illustrated in Figure

Axisymmetric contact between a frictionless conic and an elastic layer perfected bonded to rigid substrate.

The solution of the axisymmetric contact problem depicted in Figure

Polynomial fitting results of (a)

The viscoelastic behavior of materials can be simulated by the

Schematic diagram of standard solid model where a first spring (whose stiffness is

For the viscoelastic situation, both Lee and Radok [

To validate the present model, AFM-based creep tests have been performed on SMMC-7721 cell.

SMMC-7721 cells were revived after being frozen in freezer and were incubated in Roswell Park Memorial Institute- (RPMI-) 1640 media with 10% of fetal bovine serum (FBS) and antibiotics (penicillin-streptomycin solution). The protocol for the culture and fullerenol treatment of SMMC-7721 cells have been described in detail elsewhere [

The module of the AFM employed in this study is JPK NanoWizards 3 BioScience (Berlin, Germany), and it is mounted on an inverted optical microscope (Olympus IX71; Tokyo, Japan), allowing the AFM and optical microscope imaging simultaneously. The criterion for cantilever selection is that the compliance of the cantilever should be within the range of the sample compliance. For very soft and delicate cells, the softest cantilevers are available with spring constants ranging from 0.01 to 0.06 N/m (JPK Application Note). Before indentation, the spring constant of the AFM cantilever was calibrated. A silicon nitride cantilever, whose spring constant is 0.059 N/m after calibration, was used for cell-tip indentation in this work. The probe is a square pyramid tip with a half-opening angle of

Schematic of a compliant semi-infinite space indented by (a) a square pyramid and (b) a conic indenter.

Figure

The determination of viscoelastic properties of material is commonly realized by the creep response to a prescribed load. The loading method of indentation force illustrated in Figure

Schematic of the AFM indentation force versus time (a) and its approximation (b) by Heaviside step function.

Priority to creep test, the contact mode of AFM was used for topography imaging of the cells. The AFM deflection images of both control (Figure

(a) and (b) represent AFM deflection image of control and treated cells, respectively. (c) and (d) denote the 3D distribution of cell height for control and treated cells, respectively. (e) and (f) are the corresponding statistics of cell height value for control and treated cells, respectively.

The statistics results of cell height and surface roughness. Data are expressed as mean ± SEM of more than 30 cells from 3 separate experiments, where key significance values are shown,

Although the elastic modulus is frequently used to characterize the mechanical properties of biological cells, it does not present a complete description. It can be seen from the blue bold curves in Figure _{∞}, and viscosity

Fitting results of control cells ((a) and (c)) and treated cells ((b) and (d)) by the two models.

The viscoelastic parameters of control and treated cells were determined according to Sneddon’s solutions and the present model, and their mean values are presented in Figure

The statistics results of (a) _{∞}, and (c)

From Figure

In our study, we treat the cell as a homogeneous material and thus present a global equivalent quantification of viscoelastic properties of the cell. We admit that the assumption of homogeneity is a limitation in our present work, and inhomogeneous model would present more details. For example, Feneberg et al. [

In order to further validate the capability of the present model in alleviating the substrate effect, we present a test of it on different height area of cell elaborated as follows. As shown in Figure

(a) Deflection image of single control cell with the scanning range of

In this paper, we first introduce the present model based on the contact mechanics of thin film, and this model underlies the interpretation of flattened cell subjected to AFM indentation. The present model relieves the major assumption of semi-infinite space of classic Sneddon’s solutions to account for the realistic morphology of spread cells. Afterwards, the model is extended to viscoelastic constitution to reflect cell’s viscoelastic nature. The AFM-based creep test was conducted to validate the present model. The topography imaging of SMMC-7721 cell confirms that cells exhibit flattened morphology which justifies the application of the present model. The fitting results have shown that the present model can not only describe very well the creep behavior of the SMMC-7721 cell, but also avoid the overestimation of elastic and viscosity properties of thin film due to substrate effect. Hereupon, we account for the suppression of overestimation by the present model in terms of correction factor. In addition, the present model could identify the variations of the SMMC-7721 cell and its fullerenol-treated counterpart in terms of the extracted viscoelastic parameters, which reveals its instructive significance in understanding fullerenol-induced effect on the viscoelastic properties of cancerous cells and the potential in anticancer drug in terms of fullerenol application.

An abstract version of the manuscript was presented at ICBCBBE 2018: 20th International Conference on Bioinformatics, Computational Biology and Biomedical Engineering.

The authors declare that there are no conflicts of interest within the present study.

The authors are grateful for the technical support from the Laboratory of Precision Engineering and Surfaces of the University of Warwick and the International Research Centre for Nano Handling and Manufacturing, Changchun University of Science and Technology. This project has been partially funded by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant Agreement no. 644971, FP7 MCA-IRSES (612641), and the China-EU Research Programme (S2016G4501).