Materials Science and Engineering C 59 (2016) 636–642

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Materials Science and Engineering C

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Computational local stiffness analysis of biological cell: High aspect ratiosingle wall carbon nanotube tip

Amin TermehYousefi a,⁎, Samira Bagheri b, Sheida Shahnazar b, Md. Habibur Rahman c, Nahrizul Adib Kadri d

a Department of Human Intelligence Systems, Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology (Kyutech), Japanb Nanotechnology & Catalysis Research Centre (NANOCAT), IPS Building, University Malaya, 50603 Kuala Lumpur, Malaysiac Department of Computer Science and Engineering, University of Asia Pacific, Green Road, Dhaka-1215, Bangladeshd Department of Biomedical Engineering, Faculty of Engineering, University Malaya, 50603 Kuala Lumpur, Malaysia

⁎ Corresponding author.E-mail address: at.tyousefi@gmail.com (A. TermehYou

http://dx.doi.org/10.1016/j.msec.2015.10.0410928-4931/© 2015 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 6 May 2015Received in revised form 6 September 2015Accepted 14 October 2015Available online 23 October 2015

Keywords:Atomic force microscopySingle cell analysisCNTsProbing technique finite element analysisMechanical displacement of cellC-NEMS

Carbon nanotubes (CNTs) are potentially ideal tips for atomic forcemicroscopy (AFM) due to the robustmechan-ical properties, nanoscale diameter and also their ability to be functionalized by chemical and biological compo-nents at the tip ends. This contribution develops the idea of using CNTs as an AFM tip in computational analysis ofthe biological cells. The proposed software was ABAQUS 6.13 CAE/CEL provided by Dassault Systems, which is apowerful finite element (FE) tool to perform the numerical analysis and visualize the interactions between pro-posed tip and membrane of the cell. Finite element analysis employed for each section and displacement of thenodes located in the contact areawasmonitored by using an output database (ODB).Mooney–Rivlin hyperelasticmodel of the cell allows the simulation to obtain a newmethod for estimating the stiffness and spring constant ofthe cell. Stress and strain curve indicates the yield stress point which defines as a vertical stress and plan stress.Spring constant of the cell and the local stiffnesswasmeasured aswell as the applied force of CNT-AFM tip on thecontact area of the cell. This reliable integration of CNT-AFM tip process provides a new class of high performancenanoprobes for single biological cell analysis.

© 2015 Elsevier B.V. All rights reserved.

1. Introduction

Researchers have demonstrated various methods to predict theinteraction of the biological cells in bioprocesses which is due to thelack of fundamental measurement approaches in mechanical interac-tions of the cells. Nowadays, individual cell can be mechanicallymeasured using their structural entities by applying different methodssuch as: magnetic twisting cytometry [1], ball tonometry [2], micro-manipulation [3], micropipette aspiration [4], atomic force microscopy[5], optical tweezers [6], cytoindentation [7], and turgor pressureprobe [8]. Among these methods, AFM is playing a significant role indifferent fields, such as nanotechnology, materials science, surface sci-ence and biology [9,10]. AFM has been used to probe a number of prop-erties inherent to microbial cells, mammalian cells and biomoleculesincluding analysis of cellular mechanical strain and elasticity, due tothe precise application of low forces to cells with minimal disruption.One of the key parameters to broaden AFM applications is to investigatenew probe types which have better lifetime, higher resolution, andhigher mechanical property to provide quantitative and accurateanalysis of biological cells [11,12].

sefi).

Researchers have nominated carbon nanotube tips due to the severaladvantages, including (a) high aspect ratio for imaging deep and narrowfeatures, (b) low tip sample adhesion for gentle imaging, (c) the ability toelastically buckle rather than breakwhen large forces are appliedmakingthem highly robust, and (d) the potential to have resolution, 0.5 mm inthe case of individual single-wall carbon nanotubes (SWCNTs). Further-more, the wettability of the CNTs' surface is an important property,governed both by chemical composition and geometrical structure ofthe contact surface, which can play a key role in the CNTs' performanceas an AFM tip. In fact, by using CNTs as an AFM tip, the probed materialremains non-reactive [13,14].

The above characteristics make carbon nanotube to be an ideal tipfor probing in AFM. First CNT-AFM probe was developed in 1996 byTans et al. [15] which demonstrated considerable potential in probingmechanism of AFM [16]. In a pioneering study, Lieber group also [17,18] prepared CNT-AFM probes using mechanical assembly. Someother scientist conducted studies on the possibility of carbon nanotubesfor force microscopies [19]. Fig. 1 is a SEM image showing a nanotubeattached to a conventional micro-fabricated probe.

Previously we had simulated CNTs to find out the variation ofmechanical properties of CNTs while immersing Nano-particles on thesurface of SWCNTs [21,22]. But in this contribution, a novel finite ele-ment analysis of SWCNT-AFM tip is reported to obtain the local stiffnessanalysis of the cell. Available powerful finite element analysis software

Fig. 1. A carbon nanotube attached to a conventional micro-fabricated probe [20].

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ABAQUS 6.13 CAE/CEL was used to model and analyze the membraneand CNT tip. With greater understanding of the way in which mechan-ical properties of SWCNT-AFM, it may easily possible to continuouslytune the selectivity and sensitivity of nanotubes in biological applica-tions. Fig. 2 shows the schematic diagram of the proposed idea.

2. Theoretical analysis

Mechanical properties are an important determinant of stress gener-ation in any cell experience molding. To motorize the biomechanics of

Fig. 2. Schematic diagram o

Fig. 3.Meshed mod

the cell such as stiffness or spring constant, we have to determine thedisplacement of the cell by applying load on the model [23]. The stiff-ness of a cell can be described by the tensile elasticity or Young's mod-ulus (E), which is measured in units of Pa (N/m−2) and represents theratio between the applied stress on the cell (force per unit area) andthe resulting strain (fractional change in length). Spring constant alsocan be obtained by indenting technic of the cell by CNT-AFM tip [24].

3. Finite element analysis procedure

Finite element is versatile and effective distributed-parametersmodeling method, which previously has been used for variousmodeling regarding the AFM [25–27]. In the pioneering study,ABAQUS 6.13 CAE/CEl finite element toolkit was applied in order toobtain the mechanical behavior of the cell by CNT-AFM tip.

The complete model of the simulation was divided into three parts:1) CNT-AFM tip, 2) cell and 3) an underlying substrate to hold thesample tightly. Axial displacement of the CNT can be applied by acontrollable force during the specific time to monitor the deformationof the cell wall.

4. Finite element analysis of carbon nanotubes

CNTs are extremely tight and stable materials according to theirstrength sp2 hybridized covalent C–C bonds. The bonding mechanismin a carbon nanotube system is similar to that of graphite, since CNTscan be thought of as a rolled-up graphene sheet. Considering the highYoung's modulus (elastic modulus) and significant density of carbonnanotubes (approximately five times more than steel), they are excel-lent candidates for AFM probing [28]. Several modeling schemes suchas the Tersoff–Brenner potential, an empirical force-constant model,

f the SWCNT-AFM tip.

el of CNT-tip.

Table 1Geometry and mechanical properties of CNT-tip.

Geometric properties of the CNT-tip

Radius 1 nmShell thickness 0.2 nm

Mechanical properties of the CNT-tipYoung modulus 1.25 TPaPoisson's ratio 0.3Density 2.1 μg/nm3

Table 2Geometry and mechanical properties of a biological cell.

Geometric properties of the biological cell

Thickness 1 nmRadius 20 nm

Mechanical properties of the cellYoung modulus 0.0015 GPaPoisson's ratio 0.45Density 1.084 μg/nm3

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structural and continuum mechanics models, tight binding theory arepredicted physical parameters of carbon nanotubes such as elasticmodulus, tensile strength, shear modulus, energy and stress per unique[29].

According to the structural mechanics of SWCNTs, C–C bonds wereprofiled as a circular shape with 1 nm radius as shown in Fig. 3. Accord-ing to the zero potential energy of CNTs, initial atomic structure frame ofSWCNTs is in a space model. The general-purpose shell element isadopted for analysis, which provides accurate finite-strain solutions inall loading conditions regarding 0.2 nm thickness of the shell. Thesection of SWCNTs is continuum homogeneous shell with thinnessintegration [30]. The materials behaviors were choosing as an elasticmaterial with the ranging of 1.25 TPa and Poisson's ratio of 0.3 in anisotropic mode with 2.1 μg/nm3 density. Table 1 summarizes theproperties of CNT-AFM tip.

5. Finite element analysis of biological cell

Geometrical parameters of the oval cell such as radius and shellthickness will be considered while simulating the effect of geometryproperties on quantitative data obtained by nano-probing of the cellusing CNT-AFM tip. Therefore, deformation theory of the shell andbending rigidity is dependent on the finite element modeling of thecell membrane.

Researchers have reported several methods to indicate the mechan-ical properties of the different biological cells. Here we have developedthese ideas to simulate the optimumfinite element analysis of biologicalcell using numerical analysis [31]. It is well known that, in the elasticdeformation theory of shell, the bending rigidity (D), or the flexural ri-gidity, of a thin isotropic shell is proportional to the cubic power of

Fig. 4.Meshed model o

the thickness of thewall. Thus the bending rigiditymay be expressed as:

D ¼ Eh3

1−V2� � ð1Þ

where E is the Young's modulus, h is the wall thickness, and v is thePoisson ratio.

The model for cell behavior is based on a choice of constitutiverelationships (e.g. linear-elastic, Mooney–Rivlin) which describe thematerial of the cell wall and governing equations which link the consti-tutive equations to the geometry of the cell during compression. In thecase of Feng and Yang (1973) the constitutive equations were basedon a Moony–Rivlin model initially used to describe rubber. In theMooney–Rivlin model, the strain–energy function W of an isotropicincompressible material is:

W ¼ C1 I1−3� �þ C2 I2−3

� � ¼ C1½ I1−3� �þ β I2−3

� � ð2:aÞ

I1 ¼ J−2=3I1; I1 ¼ λ21 þ λ2

2 þ λ23; J ¼ det Fð Þ

I2 ¼ J−2=3I2; I2 ¼ λ21λ

22 þ λ2

2λ23 þ λ2

3λ21

ð2:bÞ

where C1 and C2 are the material constants with the dimensions ofstress β = C2/C1 for a homogeneous and isotropic, incompressibleelastic material C1 is equal to 6E. I1 and I2 are the strain invariantswhich may be expressed in terms of the principal stretch ratios in themeridional and circumferential directions of the deformed surface. F isthe deformation gradient, for example, for incompressible materialsJ = 1. For a Mooney–Rivlin material, the value of β has been taken as0.1, as suggested by Green and Adkins.

f the biological cell.

Fig. 5. FE analysis of the model.

639A. TermehYousefi et al. / Materials Science and Engineering C 59 (2016) 636–642

Although single and double-shell models have been the most com-monly models of the biological cells used for theoretical calculations,in this work we use an oval shelled model shown in Fig. 4, consistingof cytoplasm,membrane and inner and outer walls. Thin-walled spherewith a linear elastic constitutive equation for the cell wall material canrepresent the mechanics of the membrane. The high strain ratesallowed the cell wall permeability to be neglected. Thereby allowingthe initial stretch ratio of individual cells was found from compresseddata with the corresponding elastic modulus.

Geometrical parameters determined by pervious researches on cellinclude the thickness of 1 nm and external radius of 20 nm, mass

Table 3Summary of material parameters.

CNT-AFM tip Part Leng

Simulated carbon nanotube 11 n

Cell Part RadiMooney–Rivlin hyperelastic 20 n

Substrate Part WidSilicon 40 n

density of the cell (1.084 μ g/nm3), Young's modulus (0.0015 GPa)[32], and position ratio (0.45). Therefore, the total area of the ovalshape stands as A=4πr2= 4π (20 nm)2= 0.005 pm2. Table 2 summa-rizes the geometry and mechanical properties of a biological cell.

6. Assembling the parts

Cell, CNT-AFM tip and substrate were assembled together accordingthe reaction mechanism of AFM [27]. Boundary conditions applied toconduct the applied load to the cell which include the fix boundaryconditions for the nodes were located in the top of the CNT-AFM tip,

th Radius Elements Nodes

m 1 nm 4409 2977

us Area Thickness Nodesm 37.68 1 nm 12,969

th Length Elements Nodesm 40 nm 300 672

Fig. 6. Cell displacement vs. CNT-AFM tip force.

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and the gradient nodes, which located between the contact area of thecell and the tip as shown in Fig. 5. Also the substrate has zeromovementin X, Y, and Z axes to make sure that the cell does not move withexternal load from the tip. In Table 3, a summary of geometry andmesh elements of each part of the system is provided.

Fig. 7. Time independ

7. Local stiffness and spring constant measurement

Measurement of the stiffness in the case of our research depends onthe direction of the force and displacement of the cell per each element.ODB output results indicate the local stiffness of the cell by monitoring

ent applied force.

Fig. 8. a) CNT-AFM tip indention into the cell, b) cell rupture.

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the displacement of the contact area of the cell by the CNT AFM tip. Stiff-ness of the cell is measurable by the mechanical analysis method of theCNT-AFM tip as the following principle:

kcell ¼Δcell

Fð3Þ

where F is an applied force of the tip on the contact area of the cell andΔcell is the deformation of the cell. According to the output results of themodel, the relation between applied force of CNT-AFM tip and displace-ment of the cell leads to simulation. The local stiffness of the cell hasbeen simulated regardless of temperature (Fig. 6).

To obtain the progressive deformation of the cell, the applied force ofthe CNT-AFM tip was measured during the specific period (Fig. 7) toobtain the failure point of the cell while indention of the tip.

The indention of CNT-AFM tip into the cell and the outer layer of cellrupture are obvious in Fig. 8 a and b respectively. Considering the timeindependent force, we can define a normal stress by monitoring thedisplacement of the cell in the force orientation as shown in Fig. 9.

By deflation elements located in the cell's contact area and thedeflection of radius on that location which is in the same direction as

Fig. 9. Stress vs strain diagram.

the applied force, normal stress can be estimated base on the followingformula:

E ¼ stress=strain¼ NormalStressof element= cell radiusdeflection=cellradiusð Þ

The alteration of the radius after applying force was measuredthrough the Query tools of abacus which indicates 1.04764 e−001.Therefore, the normal stress and strain can be defined as 0.0349 and1.52 e−4. The hyperplastic behavior of the cell during the applied loadcreates the yield point of the stress which has demonstrated the borderof the hyperelastic behavior and elastic region of the cell.

Previous studies on themechanical properties of CNTs have revealedthat they possess high stiffness, strength and flexibility. The obtainablespring constant of the CNTs has been predicted with a range of0.001–0.05 N/m. The spring constant of the carbon nanotubes can bereduced by increasing the length or reducing the cross section area ofit. The spring constant of the cell, kcell can be calculated from therelationship of two springs in series as described in:

kcell ¼ kCNTΔtotal−Δcell

Δcell

� �ð4Þ

where kcell the spring is constant of the CNTs, Δtotal is the totaldisplacement of the CNTs in the applied force direction and the cell,and Δcell is the deformation of the cell. Performed springs constantwhich have been measured before, yielded values between 0.01 up to0.5 N/m for cell which is cover our results. Total displacement of thecell can be achieved by displacement of the nodes before and afterapplying load through the query tools of abacus.

Sneddon model for a nonlinear hyperplastic material directs us tocalculate the indentation depth while the interaction between CNTAFM tip and flat (cell) geometry is shown as:

б ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9F2 1−μ2

� �216E2a

3

sð5Þ

with μ being the Poisson's ratio, E the Young's modulus, б is theindentation depth and F is the applied force. However, this methodhas been largely developed on epithelial cells but in our case, it canalso be applicable due to lateral cell wall thickness and membranebehavior of the biological cell.

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8. Conclusion

In this contribution, we have investigated a novel method for singlecell analysis of the biological cell using carbon nanotubes as an AFM tip.The finite element model of CNTs was developed as well as the finiteelement of the cell. The results clearly demonstrate that the CNT-AFMtip is able to probe the surface of the cell to measure the stiffness of itby adjusting the material of the cell according to the nonlinearMooney–Rivlin hyperelastic material. The parameters of CNTs and cellhave been obtained from the previous experiment on the alteration ofCNT mechanical properties. By analyzing the applied load numerically,the local stiffness, spring constant, yielding point of the cell and inden-tion depth while penetration were calculated. The proposed methodmight be developed for measuring different properties of any kind ofbiological cell, especially for the cancer cell analysis to obtain physicalproperties of inside a single cell.

Conflicts of interest

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the University Malaya Research Grant(RG156-12AET).

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