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Modeling and simulation of three-dimensional manipulation of viscoelastic folded biological particles considering the nonlinear model of the cell by AFM Moharam Habibnejad Korayem , Mahsa Vaez , Zahra Rastegar PII: DOI: Reference:

S0167-6636(19)30942-1 https://doi.org/10.1016/j.mechmat.2020.103342 MECMAT 103342

To appear in:

Mechanics of Materials

Received date: Revised date: Accepted date:

30 October 2019 13 January 2020 27 January 2020

Please cite this article as: Moharam Habibnejad Korayem , Mahsa Vaez , Zahra Rastegar , Modeling and simulation of three-dimensional manipulation of viscoelastic folded biological particles considering the nonlinear model of the cell by AFM, Mechanics of Materials (2020), doi: https://doi.org/10.1016/j.mechmat.2020.103342

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Highlights

we present the analytical nonlinear cellular mechanical models that lead to the extraction of the creep function proportional to the biological particle behavior. Cylindrical and crushed cylindrical geometry considered in manipulation simulation. The cell is modeled with 2nd and 3rd order nonlinear spring and damper which are parallel and series. At the end, 2nd order nonlinear Kelvin selected as the most appropriate model. Comparison with the cell model of power-law and experimental data led to correction coefficients in models. JKR viscoelastic contact model was proposed for nanoparticles with cylindrical and crushed cylindrical geometry. The application of cell models in the contact model and subsequently modeling of the first phase of the manipulation, considering folding factor, has been done. By simulating the main motion modes, including the mode of the slip of the probe tip on the particle, particle's rotation on the surface and the mode of slipping the particle on the surface, the force and critical times were obtained.

Modeling and simulation of three-dimensional manipulation of viscoelastic folded biological particles considering the nonlinear model of the cell by AFM

Moharam Habibnejad Korayem*1, Mahsa Vaez, and Zahra Rastegar Robotic Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran .

Abstract Previous research on biological particles manipulation has taken into account linear cellular models and spherical geometry, Whereas, particles such as bacteria and cancer cells have cylindrical and crushed cylindrical geometry. On the other hand, cell behavior is nonlinear. Therefore, it is important to apply above mentioned models in the manipulation models and to investigate the modes of motion in the cylindrical nanoparticle manipulation to calculate the critical time and forces in order to understand the properties and behavior of these particles. In this paper, we present the analytical nonlinear cellular mechanical models that lead to the extraction of the creep function proportional to the biological particle behavior. Cylindrical and crushed cylindrical geometry considered in manipulation simulation. The cell is modeled with 2nd and 3rd order nonlinear spring and damper which are parallel and series. At the end, 2nd order nonlinear Kelvin selected as the most appropriate model. Comparison with the cell model of power-law and experimental data led to correction coefficients in models. Hereafter, JKR viscoelastic contact model was proposed for nanoparticles with cylindrical and crushed cylindrical geometry. Then, the application of cell models in the contact model and subsequently modeling of the first phase of the manipulation, considering folding factor, has been done. By simulating the main motion modes, including the mode of the slip of the probe tip on the particle, particle's rotation on the surface and the mode of slipping the particle on the surface, the force and critical times were obtained. According to the results, for a particle with a cylindrical geometry, the slip mode of the particle on the surface happens in 505.4 milliseconds and 5 microseconds faster than other modes. Besides, applying the folding factor causes an increase of 7% in the critical time. Because the folded shape of the cell surface causes more disturbance and friction, it requires more time and force to move the particle away and is matched to the physics of the problem. For a particle with a crushed cylindrical geometry, the slip of the probe tip on the particle occurs in 420.7 milliseconds and 10 milliseconds faster than other modes and under the force of 0.7255 micro N and about 71 percent less than the others. Also, the application of the folding factor for this particle contributes to an increase of 24% in critical force and an increase of 11% in the critical time.

*Correspondence: [email protected] 1

Robotic Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

Keywords: Biological Particles, Nonlinear Cells Model, Viscoelastic Contact Model, Folding Factor, AFM 1. Introduction Living cells, as physical entities, have physical and structural properties that enable them to withstand a physiological environment, such as mechanical stimulation that occurs inside and outside the body. Any deviation from these properties not only eliminates the integrity of the cells but also disrupts biological functions. Lim and his colleagues examined the mechanical responses of live cells to transient and dynamic loads [1]. The mechanical properties of biological cells are investigated by various methods. The most common are optical scissors, magnetic beads, and Micropipette aspiration. However, these methods cannot provide the precision of atomic force microscopy [2]. To describe the cellular behavior Desperate and his colleagues modeled cell properties with spring and damper and extract their creep function. These models often consist of one or more elastic and viscous elements that are characterized by springs and damper. Unlike simple mechanical models, many researchers have reported power-law behavior to describe cell creep function with tensile-action steps which contain a large number of the time constant [3]. Stiffness measurements are performed on the forceindentation curves that are appropriate for identifying healthy and patient cells. Models are used to interpret the widespread changes observed in cellular responses that express the mechanisms that affect cell function. Although Linear and rheological models are useful, nonlinear models are required despite complexity [4]. Korayem and colleagues developed various contact theories for biological nanoparticles with cylindrical and crushed cylindrical geometry and manipulation with AFM. They initially investigated the effective contact forces and their effect on contact mechanics. Then, the Hertz Contact Model was simulated for gold and DNA nanoparticles with spherical, cylindrical, and crushed cylindrical geometry. The contact in cylindrical and crushed cylindrical geometry was modeled based on the theories of Lundberg, Dowson, Nikpur, Hoeprich and Hertz for the manipulation of nano/microparticles [5]. Hui and Baney provided the contact model for viscoelastic spheres. Their model was somehow developed JKR model. Their numerical simulations were well matched to the experimental results of the past. However, it was difficult to determine the surface energy of matter using this theory and alternative methods should be used to determine the adhesion properties [6]. Khaksar modeled and simulated contact theories for cylindrical and crushed cylindrical geometries. For the cylindrical particles, the theorems Hertz and JKR showed good agreement with the finite element [5]. The manipulation of spherical nanoparticles was first simulated using AFM by Sitti and Hashimoto. They provided a two-dimensional dynamic model for assembling spherical micro/nanoparticles [7]. For the cylindrical particles due to the deformation of the spherical particles, frictional conditions have changed and other models have to be developed. Like spherical particles, different models for cylindrical nanoparticles are presented. Saraee first initiated the 3D manipulation of biological nanoparticle modeling in lateral and longitudinal movements to position correctly and accurately in biologic processes. For this purpose, comprehensive 3D modeling has been done to obtain displacements, speeds, and accelerations. Then, using the Newton-Euler equations, the nanoparticle’s forces and moments are obtained. Since the micro/nanoparticles studied in this study were cylindrical

micro/nanoparticles, cylindrical contact models have been introduced to accurately determine the micro/nanoparticle deformation in the 3D manipulation equations. Eventually, the forces needed to start the manipulation process and the resulting deformities have been shown. Then the dynamic behavior of the nanoparticle movement was simulated by solving equations for different motion modes and finding the force necessary to move in any dynamic mode and compared with the results of the two-dimensional modeling [8]. Hoshiar modeled and simulated dynamic modes in the manipulation of nanorods. In this research, five motion modalities including particle swirl on substrate, slip of the probe tip on the particle in the first direction, the particle rotation on the substrate, the slip of the particle on the substrate and the slip of the probe tip on the particle in the first direction were examined [9]. Sooha studied the dynamical modeling of three-dimensional nano-manipulation modes of nanoparticle with a viscoelastic contact model for spherical particles, considering cellular linear models [10]. Shahali experimentally determined the folding factor coefficient and roughness parameters for breast cancer cell to investigate their effect on the three-dimensional manipulation of the viscoelastic particle [11]. So far, in the research conducted, the contact of the AFM probe and the nanoparticle has been modeled mainly by considering the behavior of the particle's elasticity. While biologically active viscoelastic behavior occurs. On the one hand, these contact models have not been extracted for viscoelastic cylindrical and crushed cylindrical geometry. In addition, mechanical models of nonlinear biochemical behavior have not yet been investigated. The above items will be effective in calculating the force and critical time in 3D manipulation in different motion modes. Therefore, this paper represents the analytical nonlinear cellular mechanical models that lead to the extraction of the creep function proportional to the biological particle behavior. The cell is modeled with 2 nd and 3rd order nonlinear spring and damper which are parallel and series and at the end, most appropriate model will be selected. Hereafter, JKR viscoelastic contact model will be proposed for nanoparticles with cylindrical and crushed cylindrical geometry. Then, the application of cell models in the contact model and subsequently modeling of the first phase of the manipulation, considering folding factor, will been done. By simulating main motion modes including slipping of tip on particle, slipping of particle on substrate and rolling of particle on substrate critical time and force will obtain. 2. Modeling of nonlinear mechanical cellular models The living cells has its own structural and physical properties that enable them to withstand the physiological environment as well as mechanical stimuli inside and outside the body. Deviating from these properties causes the cell to malfunction and impair its function. Therefore, it is necessary to do a quantitative study in single-cell mechanics [1]. Extensive studies have been done on cellular mechanics, which suggests that the Maxwell and Kelvin models are reasonable due to viscoelastic cell behavior. Therefore, the nonlinear models of these two models have been investigated. For example, a cell membrane can be considered as a spring with stiffness E and cytoplasm with a damping coefficient C. In this paper, four nonlinear viscoelastic models are used for cells. Then, the continuum mechanics calculations for each model were done and creep functions were reported in Table (1). 2-1 2nd-order nonlinear Kelvin model solution

In the nonlinear Kelvin model, cell is modeled with a spring with the stiffness of E and a damper with c coefficient, so the stress on the cell, σ, obtained using equation (1). σ ̇

√ ∫

√

∫

(

)

(1)

√

is the constant value of the tension and t is the time. The 2nd-order Kelvin creep function ε is obtained by equation (2):

Where,

√

√

(2)

2-2 General solution of the nonlinear Kelvin model In the nonlinear Kelvin model, cells are modeled with nonlinear spring and damper, and thus, the stress on the cell can be obtained as follows (Equation 3): ̇

(3)

Which is a n-order differential equation. The solution is available for different values of n and different boundary conditions. The creep function of the 3rd Order nonlinear Kelvin Model was solved using Mathematica software. Then, to extract the creep function, Get data Digitizer software was used to fit the curve and the results were obtained. {

(4) 2-3 2nd order nonlinear Maxwell model solution

The nonlinear Maxwell is modeled with nonlinear spring and damper in series, and the strain of the cell is obtained from equation (5): ̇

̇

̇

̇

√

(5)

For constant stress, √

, at t = 0, we have:

√

(6)

2-4 General solution of the nonlinear Maxwell model In the nonlinear Maxwell model, the cell is modeled with nonlinear spring and damper, and the cell’s strain is obtained using equation (7). √ ̇

̇

√

(7) So:

√

√

(8)

Creep can be extracted for different roots. The results of extraction of nonlinear creep functions are presented in Table (1). It should be noted that in this paper, solving models in general is considered for assigning a value of 3 to the n parameter in the simulations. Table 1. Nonlinear creep function of biological cells Nonlinear model 2nd order Kelvin

Stress-strain equation

Creep function √

̇

√

3rd order Kelvin ̇

2nd order Maxwell

√

3rd order Maxwell

√

√

̇

̇

̇

̇ ̇

√

̇

√

√

3- Viscoelastic contact model for cylindrical and crushed cylindrical geometry

As we are modeling the behavior of the AFM, the extraction of force-indentation diagrams is important. So we need a force equation and indentation rate. This equation is affected by the creep function and comes from the contact models. The most suitable model is the one which most matches experimental results of force-indentation. Considering that the particles are biological, the viscoelastic contact models are more adapted to the particle’s behavior than the elastic models. Intended geometries are a cylinder and crushed cylinder. The contact surface geometry of the probe and particle is an ellipse and for contact surface of particle and substrate is a rectangle. The calculation of penetration depth is very important in the manipulation and identification of the properties of particles. In order to investigate the theory of contacting two objects in general, we assume that the main radius of the curvatures at the contact point are and in accordance with Figure 1 they are perpendicular.

Figure 1. Two bodies and their main radius contacting each other [12]

The following relations are presented for force-indentation variations in generalized coordinates in which m(t) is the oval contact radius in the contact region [12]. (9) √ (10) √ (11) Simultaneously solving of three equations, the force-indentation values in two contact surfaces will be obtained in which the convolution is defined as follows [12]: ∫ (12) So:

∫ (13) on the other hand: (14)

(15) (

)(

)

(16) According to the Hertz theory, the small and large radii of the elliptical contact region are derived from equation 17 [5]. (17) Where ν is the Poisson’s ratio, R is the radius of the object, E is the elastic modulus and indices p and t are respectively related to the particle and tip, so: (

)

(

) (18)

(19)

(20) (21) Using the elastic hertz at the highest depth, we obtain the highest m value. We can consider the variations of m(t) linearly. Finally, after the placement and simulation, the forceindentation relations are extracted for creep functions. Cell’s plasma membrane is folded and corrugated (Fig. 2). This parameter is presented as folding factor to provide more accurate modeling of contact models. Its value is obtained from equation (22) and for MCF7 cells the experimental results are reported to be 1.028. are the values of the small and large radii of the cell, a and b are values added due to the folding factor [11].

(22)

So, the actual contact radius

obtained from equation (23), in which equivalent to equation (17).

is

(23)

Figure 2. Folded and smooth surface of a spherical cell [11].

It should be noted that the amount of the folding factor coefficient is obtained by dividing the actual area of the cell surface into the ideal area and the smooth surface of the cell. For a cylindrical particle, a spherical needle with a particle of cylinders with an elliptical contact surface, and at the contact point of the cylinder and substrate, a rectangle is formed. Then, in the first step, the crushed cylinder is divided into two cylindrical and spherical sections (Fig. 3). To calculate the maximum depth of indentation of a crushed cylindrical, first equation (24) calculates the amount of force applied to the cylindrical part and the force applied to the spherical part and based on the length of the cylinder, the maximum deformation values are obtained based on cylindrical contact theories [5].

Figure 3: Schematic of particle-needle and particle-surface contact for crushed cylindrical and cylindrical geometries.

(24) In the simulation of nonlinear models, irrational graphs are obtained so the causes are discussed. Desprat and others reported nonlinear creep function in power law model in equation (25). (25)

These constants have been obtained from experiments; therefore, we compared all models with this model. Due to the constant coefficient for this model, the difference between power law and nonlinear creep function is unacceptable and requires development. Nonlinear creep functions without correction coefficients do not provide proper creep functions, but they are obtained by applying the coefficients according to power law model of creep functions in table (2). It should be noted that applying changes to linear models does not lead to changes in force-indentation diagrams. Table 2. Developed cellular creep function Nonlinear model 2nd order Kelvin

Creep function √

√

3rd order Kelvin

2nd order Maxwell

3rd order Maxwell

√

√

√

4- Manipulation and motion modes The three-dimensional manipulation free diagram of cylindrical nanoparticle is shown in Figure 4. The manipulation of micro/nanoparticles is carried out by the tip attached to the AFM cantilever probe. The AFM-based manipulation comprises two phases. In the first phase, which is the phase of particle adherence to substrate, after making sure that the tip has contacted the nanoparticle, the substrate is moved with a constant speed and in a fixed direction. The particle adhered to the surface also moves with a speed equal to that of substrate. This movement creates action/reaction forces at the tip–nanoparticle contact zone. The forces applied to the tip cause deformations (twisting, bending, and deflection) in the cantilever probe. As the movement of substrate and the adhered particle continues, the action/reaction forces between tip and nanoparticle increase and, as a result, the probe deformation and the friction force between substrate and nanoparticle increase as well. When the friction reaches its critical value, the adhesion phase comes to an end, and a further increase of force causes the nanoparticle to move on the substrate. The initial conditions of the problems along with other parameters are prepared in table 3. By considering the fixed end of cantilever as the reference point, and in view of the primary conditions of the problem, the coordinates of the free end of cantilever after its initial deflections (designated with subscript p0) are as follows: (26) (27)

(28) where L is the length of the cantilever. Due to the initial deformation of cantilever by the amount of angle , the probe tip makes a contact angle of with the particle. The kinematic equations of the position of the probe tip obtained in equations (29) to (31) at any time. In the equations, is the contacting deflection between the nanoparticle and the probe tip and parameter is the contact deformation between the nanoparticle and substrate, the values calculated in the previous section. R is contact radius, H is probe height, λ is probe tip distance from the center of mass of cylinder on the cylindrical longitudinal axis, φ is the angle of contact, θ is torsion angle and α is bending angle [8].

Figure 4. Modeling of the 3D-dimensional nanoparticle manipulation diagram [8]

(29) (

) (30) (31)

Dynamic governing equations in three directions are given by equations (32) to (34) by use of Euler-Newton equations in the center of mass of the probe’s tip. The first-order derivative of the position, velocity, and second-order derivative of the position yield acceleration. The values of the parameters, the height of probe’s tip H, the probe tip radius and the contact angle are constant, and the other values are time variables. ̇ and ̇ are also the bending and torsional angular velocity at each moment. In addition, the speed of the substrate is considered constant in both directions of x and y [8]. ̈ ̈

̈

̇ (32)

̈ ̈

̈

̇ (33)

( ̈ ̈

̇ ̈

̈ ̈

̇ ̇ ̇

) (34)

Inserted force on the nanoparticle is computed in equation (32) [8]: √

(35)

Considering the forces initiated on the nanoparticle and equilibrium equations, different motion modes are presented. According to the tests, the tip slipping on the particle, the slipping of the particle on the substrate and the rolling of particle on the substrate are three

possible modes which the equations are presented to determine the critical motion conditions [9]. The input forces from the probe to the nanoparticles and the reaction forces at the contact point (probe and particle contact T, particle and substrate contact S), contact force and friction between the substrate and nanoparticle considered as a massive force. The surface adhesion in the slipping mode of the probe on the particle , the friction coefficient for the probe slip on the particle , the contact surface of the probe and the particle , the surface adhesion in the slip mode on the substrate , the friction coefficient for the slip of particle on substrate , area of the particle and substrate contact , the surface adhesion in the rolling mode of the particle on the substrate and , friction coefficient for particle rolling on and , are effective parameters in motion modes. Equations (36) to (38) are presented to determine the critical conditions of motion [9].

1-slipping of probe on particle

(36) 2-slipping of particle on substrate

(37) 3-Rolling of particle on substrate

(38) Where:

(39) (40) 5- Results of the manipulation simulation with linear and nonlinear models In this simulation, the cell is considered as viscoelastic and the tip of the probe as an elastic material. The properties of the MCF-7 and probe are given in Table 3. Table 3: constant quantities [10], [14]. Parameter

quantity

Parameter

Quantity

60 ̈

0

90

̈

0

̇

100( nm / s )

xsub 0

0

̇

100( nm / s )

ysub 0

0

̇

zsub 0

0

̇

X

0.8 ( m)

̇ ̇

28 ( MPa)

Lp

1 ( m)

0.8

sub

0.43

p

3 (mJ / m )

Esub

69 (GPa )

Ep

913.2 ( Pa)

p

0.499

Gtip

66.54 GPa

tip

0.27

Etip

169 GPa

Htip

12 m

ttip

1 m

wtip

48 m

Ltip

225 m

R

25 (nm)

2

The force-indentation diagram is shown in Figure 5 for a particle with cylindrical and crushed cylindrical geometry. It is observed that the 2nd order nonlinear Kelvin modulus is more in line with the experimental results [16], indicating an increase in indentation with force increasing. The 3rd order non-linear Kelvin model also has an absurd output, which provides two values in force for a given indentation. Non-linear Maxwell models of order 2 and 3, which have their own spring and damper arrangement, are not suitable models due to the prediction of negative force and the lack of matching with reality.

Figure 5. Response of different mechanical biological cell models to simulation of viscoelastic mode for cylindrical and crushed cylindrical particles

In this section, the force-indentation diagram shown in Figure 6 is presented with and without applying the folding factor coefficient for the MCF-10 cell with two cylindrical and cylindrical geometries.

Figure 6. Simulations of force-indentation with and without applying the folding factor coefficient

The results indicate that we will have greater penetration by applying a folding factor for a given amount of force and an increase in the contact radius. For example, it is observed that for a depth of 0.8 micrometers, considering folding at the cell surface, the required force is reduced by about 50%. On the other hand according to the physics of the problem, if there is folding on the surface of the cell due to the shell's looseness, comparing body is completely stretched and not folded, under less force we will have more depth of indentation. Thus, the results of Fig. 4 also correspond to the problem physics. In order to verify the results, we compared the results obtained for the viscoelastic state with the experimental results, as well as the results of the simulation of the elastic state and the viscoelastic JKR according to Fig. 5.

Figure 7. Comparison the results of force-indentation of particles with cylindrical and crushed cylindrical geometry

The elastic curve is under the viscoelastic curves and its great difference from the experimental results show that the elastic model predicts less force for indentation than viscoelastic models, which illuminates the importance of providing viscoelastic models. Experimental data from reference [15] for MCF-7 breast cancer which have similar sphere geometry, have been extracted for validation purposes. The values of experimental results and results for cylindrical and crushed cylindrical geometry closeness proximity in forceindentation diagrams can indicate the validity of the simulation. As shown in the figure in order to have a certain amount of indentation depth a viscoelastic spherical, crushed cylinder and cylinder have the least force. On the other hand, in order to verify the results of Fig. 8 by reducing the length of the cylindrical part of the crushed cylinder, the results tend from the cylinder to sphere.

Figure 8. Comparison of the results of force-indentation of particles with cylindrical and crushed cylindrical geometry

In order to simulate the manipulation, we first need to determine the initial conditions and constants as described in [8] and [10]. In this simulation, the particle is considered as a viscoelastic body whose variation is varied in time. The probe and the substrate are considered elastic according to the high elastic modulus compared to the biological particle. The simulation results of the three-dimensional manipulation modes of biological nanoparticles for two cylindrical and crushed cylindrical geometries presented in Table 4. The simulation results indicate that applying folding factor in cylinder geometry for slipping of the tip on the particle causes a 27% increase in critical force and a 2% increase in the critical time. To slip the particle on the substrate, a 7% increase in a critical time and for particle rolling on the substrate will have a 2% increase in strength and a 7% increase in time. The folded surface of the cell creates more friction with contact surfaces, which requires more time and force for displacement.

Applying the folding factor for a particle with a crushed cylindrical geometry for slipping of the probe on the particle causes a 24% increase in critical force and 11% increase in a critical time. To slip the particle on the substrate, it will be 8% higher in critical times and 1% increase in time for the particle to roll on the substrate. Table 4. Critical force and critical values in different modes of 3D manipulation for cylindrical and crushed cylindrical particles With folding

Crushed cylinder

Cylindrical

Geometry

Without folding

Motion mode

Critical time (ms)

Critical force (µN)

Critical time (ms)

Critical force (µN)

Slipping tip of probe on particle

563.3

1.3803

510.5

1.0815

Slipping particle on substrate

540.5

0.8359

505.4

0.9588

Rolling particle on substrate

552.9

1.1295

511.2

1.1010

Slipping tip of probe on particle

468.5

0.9052

420.7

0.7255

Slipping particle on substrate

474.4

1.0843

437.8

1.2435

Rolling particle on substrate

475.0

1.1024

433.7

1.1185

In order to check the accuracy of the extraction results in terms of force and critical time, according reported values for the elastic sphere, the viscoelastic sphere [10], Viscoelastic sphere with folding factor [8], elastic cylinder, a viscoelastic cylinder with (out) folding factor, a viscoelastic crushed cylinder with (out) folding factor comparison in three motion modes is presented by rod graphs of Figures (9) and (10).

Critical Force

6 5 4 3 2 1 0

slipping of probe on particle slipping of particle on substrate Rolling of particle on substrate

Figure 9. Critical Force Comparison Chart in three modes of motion

According to the proximity of the range of critical force values for the viscoelastic cylindrical and crushed cylinders with elastic/viscoelastic sphere and elastic cylinder, it can be said that the extracted quantities for the modeling and simulation changes performed within a reasonable range. As shown in Fig. 9, values in elastic and viscoelastic states are very close, but it should be noted that the values usually in the viscoelastic state are below the elastic state. The reason for this is the term of loss in the viscoelastic state, as it causes a loss of some force for particle manipulation. In the same time, viscoelastic indentation acceleration is less. According to the kinematic equations, since the indentation acceleration has a decreasing effect on accelerations, it will be less in elastic than the viscoelastic mode, and therefore, the amount of force in elastic mode will be greater than viscoelastic. In addition, generally, more time or force is necessary to move a particle with cylinder geometry than a spherical particle. For a viscoelastic cylinder, the slipping mode of the particle on the substrate occurs under a smaller force than other modes of motion. For a viscoelastic crushed cylinder, the slipping of the tip on particle takes place under a smaller force than other motion modes. For a viscoelastic sphere, the slipping mode of the particle on the substrate happens under smaller force than other modes of motion. The rolling mode of the particle on the substrate for a folded viscoelastic crushed cylinder, an elastic cylinder and a folded viscoelastic sphere occur with more force than other modes. The particle slipping mode on substrate occurs for the elastic sphere, viscoelastic sphere, viscoelastic cured cylinders under more force than others. The probe tip slipping mode on a particle for a folded viscoelastic cylinder is more forceful than other modes.

Critical Time

600 400 200 0

slipping of probe on particle slipping of particle on substrate Rolling of particle on substrate

Figure 10. Comparison of critical times in three modes of motion

In Fig. 10, the results indicate that the rolling mode of the particle on the substrate occurs for the elastic cylinder and elastic sphere sooner than the other modes. Particle slipping on the substrate mode for a folded viscoelastic cylinder occurs with friction earlier than other modes. The probe tip slipping mode on a particle for a viscoelastic crushed cylinder with (out) folding factor and viscoelastic sphere with (out) folding factor happens earlier than other modes. It is observed that the behavior of the cylindrical model in the elastic and viscoelastic state has a difference in the numerical value and the order in which the motion modes occur, which leads to a better understanding of the movement of the particle under the manipulation and indicates the importance of the project. In addition, the folding factor increases the critical time for the manipulation. 6. Conclusion In this paper, after reviewing the studies, the problem of nonlinear cellular model and its solution for the 2nd and 3rd order non-linear Kelvin cell model and the 2nd and 3rd order Maxwell non-linear model were presented. Additionally, the contact model and the forceindentation diagrams expressed for particles with viscoelastic properties. In the simulation cell assumed viscoelastic (MCF-7 particle properties) and probe considered elastic. Four cell models were introduced in the proposed contact model. It should be noted that according to the experimental results and comparison with the creep function of the nonlinear model power-law, correction coefficients for creep functions applied. The most suitable cellular model (2nd order Kelvin developed model) and the most comprehensive viscoelastic contact model (JKR), the first phase of 3D manipulation was modeled. Dynamic and kinematic modeling of motion of the particle on the substrate by probe presented. Afterwards three

motion modes were modeled: slipping tip of the probe on the particle, rolling of the particle on the substrate, slipping the particle on the substrate. In the next step, the simulation of the first phase of manipulation and motion modes was performed to extract the critical time and force values. Manipulation force and motion modes diagrams plotted for the micro/nanoparticle with mentioned geometry considering folding factor. It should be noted that the force and critical time needed to start moving in any mode comes from the intersection of the friction force enters the nanoparticle in that mode, and the force entailed on the nanoparticle by the tip of the probe. According to the results, for a particle with a cylindrical geometry, the slip mode of the particle on the surface happens in 505.4 milliseconds and 5 microseconds faster than other modes. Besides, applying the folding factor causes an increase of 7% in the critical time. Because the folded shape of the cell surface causes more disturbance and friction, it requires more time and force to move the particle away and is matched to the physics of the problem. For a particle with a crushed cylindrical geometry, the slip of the probe tip on the particle occurs in 420.7 milliseconds and 10 milliseconds faster than other modes and under the force of 0.7255 micro N and about 71 percent less than the others. Also, the application of the folding factor for this particle contributes to an increase of 24% in critical force and an increase of 11% in the critical time. Furthermore, modeling in the biological liquid environment, the development of other contact models such as Hertz, Tatara, DMT, applying folding factor at the substrate and probe, second phase of manipulation, the experimental extraction of properties of other biological particles and experiments on cylindrical biological nanoparticles during the manipulation by atomic force microscope lead to extracting actual amounts of force and critical time in order to improve models and assisting in the diagnosis and treatment which was not done in this article due to time limitation.

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Please check the following as appropriate: All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.

Declaration: Ethics approval and consent to participate: Not applicable Consent for publication: Not applicable Availability of data and materials: Not applicable Competing interests: The authors declare that they have no competing interests Funding: There is no funding source Authors' contributions: Korayem supervised the project, Vaez extracted the equations and did coding, Rastegar provided the idea and the method and analyzed the data Acknowledgements: Not applicable

References 1-Lim C.T., Zhou E.H, Quek S.T., "Mechanical models for living cells", Journal of biomechanics, Vol.39, No.2, 195-216, 2004. 2-Alonso J.L., Goldman W.H., "Feeling the forces: atomic force microscopy in cell biology", Life science, Vol.72, 2553–2560, 2003. 3-Desprat N., Richert A., Simeon J., Asnacios A., "Creep function of a single living cell", Biophysics journal, Vol. 88, 2224–2233, 2005. 4-Haase K., Pelling A.E., "Investigating cell mechanics with atomic force microscopy", Journal of the royal society interface, Vol.12, 20140970, 2015.a 5-Korayem M.H., Khaksar H., Taheri M., "Modeling of contact theories for the manipulation of biological micro/nanoparticles in the form of circular crowned rollers based on the atomic force microscope", Journal of applied physics, Vol.114, 183715,1-13, 2013. 6-Hui C.Y., Baney J.M., “Contact mechanics and adhesion of viscoelastic spheres”, Langmuir, Vol.14, 6570-657, 1998. 7-Sitti M., Hashimoto H., "Controlled pushing of nanoparticles modeling and experiments", Transactions on mechatronics, Vol.5, No.2, 199-211, 2000. 8-Korayem M.H., Mahmoodi Z., Taheri M., Saraee M.B., “Three-dimensional modeling and simulation of the AFM-based manipulation of spherical biological micro/nanoparticles with the consideration of contact mechanics theories”, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, Vol.229, Issue 4, 370-382, 2015. 9-Korayem M.H., Hoshiar A. K., “3D kinematics of cylindrical nanoparticle manipulation by an atomic force microscope based nanorobot”, Scientia iranica, Vol. 21, Issue 6, 1907-1919, 2014. 10-Korayem M. H., Sooha Y. H., Rastegar Z., “Modeling and simulation of viscoelastic biological particles’ 3D manipulation using atomic force microscopy”, Applied physics A, Vol.124, 1-13, 2018. 11-Korayem M.H., Shahali S, Rastegar Z, “Experimental determination of folding factor of benign breast cancer cell (MCF10A) and its effect on contact models and 3D manipulation of biological particles”, Biomechanics and modeling in mechanobiology, Vol.17, Issue 3 , 745761, 2018. 12-Korayem M.H., Rastegar Z., “Development of rough viscoelastic contact theories and manipulation by AFM for biological particles: any geometry for particle and asperities”, in press, Applied physics A, 125:404, June 2019. 13-Desprat N., Richert A., ”Creep function of a single living cell”, Biophysical journal, Vol.88, No.3, 2224–2233, 2005. 14-Q.S. Li, G.Y.H. Lee, C.N. Ong, C.T. Lim, "AFM indentation study of breast cancer cells”, Biochemical and biophysical research communications, Vol.374, 609–613, 2008.

15-Yun C., Norde W., van der Mei H.C., Busscher .H,” Bacterial cell surface deformation under external loading”, MBio, Vol.3, No.6, 00378-12, 2012. 16. Korayem M.H., Sooha Y. H, Rastegar Z., “MCF-7 cancer cell apparent properties and viscoelastic characteristics measurement using AFM “, Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol.40, No. 297, 1-11, 2018

Author 1: Moharam Habibnejad Korayem

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Author 2: Mahsa Vaez

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